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Noname manuscript No.(will be inserted by the editor)
New Multi-implicit space-time spectral element methods1
for advection-diffusion-reaction problems2
Chaoxu Pei · Mark Sussman ·3
M. Yousuff Hussaini4
5
Received: date / Accepted: date6
Abstract Novel multi-implicit space-time spectral element methods are de-7
scribed for approximating solutions to advection-diffusion-reaction problems8
characterized by multiple time scales. The new methods are spectrally accurate9
in space and time and they are designed to be easy to implement and robust.10
In other words, given an existing stable low order operator split method for11
approximating solutions to PDEs exhibiting multiple scales, the algorithms12
described in this article enable one to easily extend a low order method to be13
a robust space-time spectrally accurate method. In space, two spectrally accu-14
rate advective flux reconstructions are proposed: extended element-wise flux15
reconstruction and non-extended element-wise flux reconstruction. In time, for16
the Hyperbolic term(s), a low-order explicit I-stable building block time inte-17
gration scheme is introduced in order to obtain a stable and efficient building18
block for the spectrally accurate space-time scheme. In this article, multiple19
spectrally accurate space discretization strategies, and multiple spectrally ac-20
curate time discretization strategies are compared to one another. It is found21
that all methods described are spectrally accurate with each method having22
distinguishing properties.23
Keywords Space-time · Operator splitting · Coupling strategy · Multiple24
time scales · Spectral accuracy25
Mathematics Subject Classification (2000) 65B05 · 65M7026
C. PeiDepartment of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]
M. Sussman?
Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]
M. Yousuff HussainiDepartment of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]
2 Chaoxu Pei et al.
1 Introduction1
In this article, novel space-time spectrally accurate numerical methods are2
presented for approximating solutions to time dependent partial differential3
equations exhibiting multiple temporal scales. Our new methods are distinct4
in that they are demonstrated to be space-time spectrally accurate, easy to5
implement, and result in methods which are robust to varying initial and6
boundary conditions and varying stiffness of source terms.7
8
Some applications in which one solves a multiple time scales partial differ-9
ential equation are combustion [6] and the transport of air pollutants [23].10
11
Many of the existing numerical methods that approximate solutions to mul-12
tiple time scales problems can be classified as either “divide and conquer”[37,13
8,38,18,24,7,20,4,6,31,32,30,19,2,28,14,15,25] or “monolithic”[21,39,36,35,14
34,33,11].15
Consider a time dependent partial differential equation of the following16
form:17
∂w
∂t= F1(w) + F2(w) + . . .+ Fm(w). (1)
A “Monolithic” method will simultaneously integrate in time all the terms18
in (1). These “Monolithic” methods [21,39,36,35,34,33,11] are space-time19
spectrally accurate and there are no splitting errors, but the performance of a20
“monolithic” method relies on the resulting large sparse system of equations21
not being ill-conditioned. For many time dependent partial differential equa-22
tions, there is no guarantee that a “monolithic method” will not break down23
(the sparse system solver will not diverge).24
25
A “divide and conquer” method sequentially integrates in time each force26
term in (1),27
∂w
∂t= Fj(w), j = 1, . . . ,m, (2)
where distinct (e.g. explicit vs implicit) time integration schemes are imple-28
mented for each distinct force term Fj , j = 1, . . . ,m. For the “divide and29
conquer” algorithm, the practitioner can pick and choose the optimal time in-30
tegration approach for a given force term Fj . “Divide and conquer” methods31
overcome the need to solve large sparse, ill-conditioned matrix systems, but32
these methods are less accurate than “monolithic” approaches. The low order33
“Divide and conquer” methods will incur splitting errors.34
35
A semi implicit spectral deferred correction (SISDC) method [30,19,2] or36
a multi-implicit spectral deferred correction (MISDC) method [6,31,32] is a37
“divide and conquer” method that is derived from a given low order “building38
block” operator split method. The (SISDC) or (MISDC) methods enable one39
Multi-implicit space-time spectral element method 3
to derive a new operator split temporal integration scheme of arbitrary order1
from a low order operator split method. In contrast to “monolithic” methods,2
(SISDC) and (MISDC) methods do not necessitate that one solves a possi-3
bly ill-conditioned sparse system of equations. It is reported in the (MISDC)4
literature [31,32], that as one increases the temporal order of accuracy, the5
splitting errors are decreased.6
7
Inspired by the work in [31,32], we have developed novel multi-implicit8
space-time spectral element methods for solving problems associated with two9
or more processes with differing characteristic time scales. The key distinction10
between the present work and [31,32] is that not only can one arbitrarily select11
the temporal order of accuracy with our new methods, but also one can select12
an arbitrarily high spatial order of accuracy. In this article we demonstrate13
spectral convergence for increasing temporal and spatial order of accuracy.14
We have developed a space-time spectrally accurate “divide and conquer”15
method which has the advantages of both “monolithic” and “divide and con-16
quer” methods (no splitting errors, high order, robust, and easy to implement),17
without having any of the disadvantages (splitting errors, ill-conditioned sparse18
systems, difficult to implement).19
2 Problem formulation20
We consider the following problem that exhibits multiple time scales
wt +∇ · (uw) = ∇ · (D∇w) + fr(w), x ∈ Ω ⊂ R2, (3)
where u = (u, v)> is the velocity vector, D is a symmetric matrix of diffusion21
coefficients, and fr(w) is the reaction term which could be stiff.22
3 Overview of our new space-time spectral element method for the23
advection-diffusion-reaction problem24
For the spatial discretization of ∇ · (uw) found in (3), we have developed25
two spectrally accurate methods for the discretization of the advective fluxes:26
(i) extended element-wise flux reconstruction (EEFR) and (ii) non-extended27
element-wise flux reconstruction (NEEFR). In the extended element-wise flux28
reconstruction, the Extended Gauss points, constructed by adding 2 points29
from the neighbors to the existing Gauss points in one spatial element (see30
Fig. 5), become the interpolation stencil. For the non-extended element-wise31
flux reconstruction algorithm, only the existing Gauss points in one spatial32
element are employed as the interpolation stencil. The idea of reconstructing33
fluxes with an extended stencil has been reported in an article by Dumbser et34
al. [10]. Dumbser et al.’s method was called the discontinuous Galerkin PnPm35
scheme (m ≥ n), where Pn represents a DG solution as a piecewise polynomial36
of degree of n, and Pm is a reconstructed polynomial solution of degree of m37
4 Chaoxu Pei et al.
that is used to compute the fluxes. Later, Luo et al. [29] proposed an improved1
scheme that is referred to as the reconstructed discontinuous Galerkin method.2
Distinct from the previous work in [10,29] that involve all information from the3
neighboring elements, our extended element-wise flux reconstruction scheme4
includes only one point from its neighbors, and thus it is compact and simple.5
6
In order for the overall (MISDC) temporal integration method to be spec-7
trally accurate, it is necessary that the building block temporal integration8
scheme(s) be at least first order accurate. In our spectrally accurate method,9
the diffusion and reaction source terms in (3) are discretized in time with10
the first order implicit backwards Euler method for the low order building11
block. A first order, explicit, I-stable building block time integration scheme12
is introduced as an explicit building block for the advection term in order13
to obtain a stable, accurate and efficient space-time scheme. The first order14
I-stable building block time integration scheme is a two-stage scheme. High or-15
der I-stable time integration scheme can be found in Bao and Jin [3], e.g., 3rd16
order Runge-Kutta method and 4th order Runge-Kutta method. We also refer17
the reader to [12,13] and the references therein for high order strong stability18
preserving time discretization methods which would also be good candidates19
for the advective temporal building block. The key principle of a spectral de-20
ferred correction (SDC) method is to construct an arbitrary high-order time21
scheme from a low-order time scheme by a series of deferred correction steps.22
In the present multi-implicit spectral deferred correction coupling method, the23
solutions obtained by the temporally low order operator splitting method are24
used as provisional solutions, and then such provisional solutions are corrected25
by a series of deferred correction steps. Compared to standard low order oper-26
ator splitting methods, the splitting error in the MISDC method is eliminated27
by exploiting an iterative coupling strategy in the deferred correction proce-28
dure. Two different spectrally accurate space-time coupling strategies for the29
MISDC time integration scheme are investigated. Also since the low order I-30
stable building block time integration scheme is a two-stage scheme, this gives31
rise to two more combinations of correction strategies.32
33
In all, there are a total of eight combinations of spectrally accurate space-34
time discretization methods that we have developed and analyzed in this ar-35
ticle. All of the methods are space-time spectrally accurate, but each method36
has advantages and disadvantages. For example, the EEFR method is more37
accurate than the NEEFR method, but the EEFR method is not extendable38
to unstructured grids. The (MISDC) correction strategy in (24) through (26)39
is expected to be more accurate than the correction strategy in (27a) through40
(27c), but more “intrusive.”41
Multi-implicit space-time spectral element method 5
4 Space-time discretization1
In a space-time discretization, we introduce a space-time domain E as E =2
Ω × [t0, T ]. A point in the space-time domain, x ∈ E , has coordinates (x, t).3
First, we partition the time interval [t0, T ] uniformly by the time levels 0 =4
t0 < t1 < ... < tE(t)
= T . The space-time domain E is then divided into5
E(t) space-time slabs. The n-th space-time slab is denoted as En = E ∩ In,6
where In = [tn, tn+1] is the n-th time interval with length ∆t = tn+1 − tn.7
Next, we divide the spatial domain Ω into E(x)×E(y) non-overlapping spatial8
elements. Let Ωne and Ωn+1e be the spatial element e at time level tn and tn+1,9
respectively. A space-time element Kne is then obtained by connecting Ωne and10
Ωn+1e . The tessellation of the space-time domain is denoted as Th.11
In each space-time slab En, the time interval In = [tn, tn+1] is divided12
into p(t) subintervals by choosing Legendre Gauss-Lobatto points tnm for m =13
0, ..., p(t), that is, tn = tn0 < tn1 < ... < tnp(t)
= tn+1. In the following, the14
length ∆t (∆t = tn+1 − tn) is referred to as a time step size while ∆tm15
(∆tm = tnm+1−tnm) is referred to as a time substep size. Fig. 1 is an illustration16
of a space-time slab.
-
6
x
y
t
tn0 = tn
tn1
tn2
tn3
tn4 = tn+1
∆t1 = tn2 − tn1
6
?
6
?
∆t = tn+1 − tnEn
Ωne
Ωn+1e
W
Fig. 1 An illustration of a space-time slab En. The time interval [tn, tn+1] is divided into 4subintervals by choosing Legendre Gauss-Lobatto points tnm for m = 0, ..., 4. ∆t = tn+1− tnis referred to as a time step size while ∆tm = tnm+1 − tnm is referred to as a time substepsize. The small rectangle cube is the space-time element Kn
e constructed by connecting Ωne
and Ωn+1e .
17
6 Chaoxu Pei et al.
4.1 Basis functions1
A discontinuous spectral element method in the collocation form [17,1] is used2
as the spatial discretization in a staggered grid. The approximate solution lo-3
cated at the Gauss-Gauss nodes, and the velocity vector defined on the marker-4
and-cell (MAC) grid. The MAC grid consist of Gauss-Lobatto–Gauss points5
for the velocity component u in the x-direction and Gauss–Gauss-Lobatto6
points for the velocity component v in the y-direction. The locations of the7
Gauss-Gauss points and the MAC grid points are illustrated in Fig. 2.
Fig. 2 An illustration of the Gauss-Gauss nodes and the MAC grid nodes in the spatialelement Ωe. Closed circles: Gauss-Gauss points for the approximate solution w. The velocityu = (u, v)> is on the MAC grid nodes, that is, open square: Gauss-Lobatto–Gauss points forvelocity component u, closed square: Gauss–Gauss-Lobatto points for velocity componentv.
8
The basis functions are Lagrange interpolation polynomials defined as fol-lows,
`gi (s) =
r∏k=0k 6=i
s− sgksgi − s
gk
, `gli (s) =
r+1∏k=0k 6=i
s− sglksgli − s
glk
, (4)
where sgi i=0,...,r are the roots of the (r + 1)-th order Legendre Gauss poly-
nomials, and sgli i=0,...,r+1 are the roots of the (r + 2)-th order LegendreGauss-Lobatto polynomials. Then, the discrete solution and the discrete ve-
Multi-implicit space-time spectral element method 7
locity vector in the space-time slab En are defined as follows,
wh(x, t)|Ωe =
p(x),p(y),p(t)∑i,j,k=0
wi,j,kφgi (x)φgj (y)φglk (t), (5a)
uh(x, t)|Ωe =
p(x),p(y),p(t)∑i,j,k=0
ui,j,kφgli (x)φgj (y)φglk (t), (5b)
vh(x, t)|Ωe =
p(x),p(y),p(t)∑i,j,k=0
vi,j,kφgi (x)φglj (y)φglk (t), (5c)
where
φgi (x) = `gi (s), with x = xa +∆x(sg + 1)/2, (6a)
φgli (x) = `gli (s), with x = xa +∆x(sgl + 1)/2, (6b)
φgj (y) = `gj (s), with y = ya +∆y(sg + 1)/2, (6c)
φglj (y) = `glj (s), with y = ya +∆y(sgl + 1)/2, (6d)
φglk (t) = `glk (s), with t = tn +∆t(sgl + 1)/2. (6e)
Here xa (or ya) denotes the lower bound of the spatial element Ωe in the x-1
direction (or y-direction), and ∆x (or ∆y) is the length of the spatial element2
Ωe in the x-direction (or y-direction).3
5 Multi-implicit space-time spectral element method4
Multi-implicit space-time spectral element methods are proposed for approx-5
imating the solutions of PDEs that have multiple source terms and multiple6
time scales, i.e., advection-diffusion-reaction problems. In this section, two new7
spectrally accurate spatial discretization algorithms and accompanying two8
different coupling strategies in the multi-implicit spectral deferred correction9
method are developed. Moreover, the efficiency of two correction strategies for10
integrating the hyperbolic force term are discussed.11
5.1 Discontinuous spectral element method in space12
We use a discontinuous spectral element method in the collocation form [17,1]13
for the spatial discretization. First, we describe the spatial discretization of the14
viscous term ∇· (D∇w) in the model equation (3) by two steps: discretization15
of the gradient operator and discretization of the divergence operator.16
– Discretization of the gradient operator. In each spatial element, the ap-17
proximate solution is located at the Gauss-Gauss points (see Fig. 2). The18
x derivative of the gradient, ∂∂x , is located at the Gauss-Lobatto–Gauss19
8 Chaoxu Pei et al.
points, and the y derivative of the gradient, ∂∂y , is located at the Gauss–1
Gauss-Lobatto points (see the MAC grid points in Fig. 2). Since the pro-2
cedures of computing the x derivative and the y derivative are the same,3
we describe the approximation of the x derivative as an example. We en-4
hance the approximate solution stencil in a spatial element, which are5
referred to as Extended-Gauss points, by adding 2 points to the existing6
p(x) Gauss points (see the Extended-Gauss points in Fig. 3). The solution7
values are then interpolated from the set of Extended-Gauss points onto8
the Gauss-Lobatto points of (p(x) + 2) that has the same number points of9
Extended-Gauss points. The gradient values on the Gauss-Lobatto points10
of (p(x) + 1) are obtained by differentiating the solution interpolated from11
the (p(x) + 2) Gauss-Lobatto points and then evaluating at the (p(x) + 1)12
Gauss-Lobatto points. An illustration of nodes used in computing the x13
derivative is displayed in Fig. 3. For an element adjacent to the domain14
boundary, the set of Extended-Gauss points are constructed by adding the15
point on the domain boundary and the point from the neighbor element16
to enhance the stencil, see Fig. 4.
Fig. 3 An illustration of nodes used in computing the x derivative in the x-direction.Extended-Gauss points (p(x) + 2) denotes the Gauss points together with two additionalpoints from the neighbors, Gauss-Lobatto points (p(x)+2) denotes the Gauss-Lobatto pointswith the same number points of Extended-Gauss, and Gauss-Lobatto points (p(x) + 1) de-notes the MAC grid points in the x-direction.
17
– Discretization of the divergence operator. In each spatial element, the diver-18
gence operator is approximated at the Gauss-Gauss points which coincides19
with the location of the approximate solution (see Fig. 2). Due to the dis-20
continuity of the gradient values at inter-element boundaries, the viscous21
flux is double-valued across element boundaries. In order to define a unique22
flux at inter-element boundaries, we replace the double valued flux with23
the average of the coincident gradient values coming from each side of the24
inter-element face. The product of the continuous gradient values ∇w and25
the viscous coefficients D on the MAC grid points are used to compute26
Multi-implicit space-time spectral element method 9
Fig. 4 An illustration of nodes used in computing the x derivative in the x-direction for theelement near the wall. Extended-Gauss points (p(x) + 2) denotes the Gauss points plus thepoint on the wall and a point from the neighbor, Gauss-Lobatto points (p(x)+2) denotes theGauss-Lobatto points with the same number points of Extended-Gauss, and Gauss-Lobattopoints (p(x) + 1) denotes the MAC grid points in the x-direction.
the differentiation. The values of the divergence operator are obtained by1
differentiating the product on the MAC grid points and then evaluating at2
the Gauss-Gauss points.3
For the spatial discretization of the advective term ∇ · (uw), we present4
two approaches for reconstructing the advective flux: extended element-wise5
flux reconstruction and non-extended element-wise flux reconstruction. These6
two approaches for computing the x derivative are illustrated in Fig. 5 and are7
explained in details as follows.8
– Extended element-wise flux reconstruction (EEFR). To reconstruct the ad-9
vective flux on the MAC grid points, the values of the approximate solution10
need to be interpolated from the Gauss-Gauss points onto the MAC grid11
points. Take the x-component for example. We enhance the approximate12
solution stencil in a spatial element, which are referred to as Extended-13
Gauss points, by adding 2 points to the existing p(x) Gauss points (see the14
Extended-Gauss points in Fig. 5). The solution values are then interpolated15
from the set of Extended-Gauss points of (p(x) +2) onto the Gauss-Lobatto16
points of (p(x) + 1). Due to the discontinuity of the solution values, the17
advective flux values are double-valued across the inter-element faces. The18
endpoints are then overwritten by either the average of the solution values19
on both sides of the inter-element faces or the upwind values at the inter-20
element faces. The values of the derivative are obtained by differentiating21
the solution values on the (p(x) + 1) Gauss-Lobatto points and then evalu-22
ating at the p(x) Gauss points. Since the method uses the Extended-Gauss23
points, it is referred to as extended element-wise flux reconstruction.24
– Non-extended element-wise flux reconstruction (NEEFR). In each spatial25
element, the values of the approximate solution on the Gauss-Gauss points26
are interpolated directly onto the MAC grid points. The endpoints, where27
the flux values are double-valued, are then overwritten by either the average28
10 Chaoxu Pei et al.
of the solution values on either side of the inter-element faces or the upwind1
values at the inter-element faces. The values of the derivatives are obtained2
by differentiating the solution values on the MAC grid points and then3
evaluating at the Gauss-Gauss points (see Fig. 5). As it uses the Gauss4
points in one spatial element, this method is referred to as non-extended5
element-wise flux reconstruction.6
Fig. 5 An illustration of two advective flux reconstructions, extended element-wise flux re-construction and non-extended element-wise flux reconstruction, of computing the x deriva-tive in the x-direction. Extended-Gauss points (p(x) + 2) denotes the Gauss points plus twoadditional points from the neighbors, Gauss-Lobatto points (p(x) + 2) denotes the Gauss-Lobatto points with the same number points of Extended-Gauss, and Gauss-Lobatto points(p(x) + 1) denotes the MAC grid points in the x-direction.
To obtain the space-time spectral element discretization, we substitute theapproximate solution wh in Eq. (5a) and the velocity vector uh = (uh, vh)>
in Eqs. (5b) and (5c) into the model equation (3), and then impose zero resid-ual at Gauss-Gauss points. After applying spectral element discretizations inspace, we obtain a system of ODEs as follows
p(t)∑k=0
wi,j,k(φglk )′(t) = (wi,j)t = FA(wi,j , t) + FD(wi,j , t) + FR(wi,j , t), (7)
i = 0, ..., p(x), j = 0, ..., p(y),
where FA(wi,j , t) is defined as follows for the extended element-wise flux re-construction,
FA(wi,j , t) =−p(t)∑k=0
p(x)+1∑ii=0
uii,j,k2wuii,j,k(φglii )
′(xi)φglk (t)
−p(t)∑k=0
p(y)+1∑jj=0
vi,jj,k2wvi,jj,k(φgljj)
′(yj)φglk (t). (8)
Multi-implicit space-time spectral element method 11
with
2wuii,j,k =
p(y),p(t)∑j′,k′=0
p(x)+1∑i′=−1
wi′,j′,k′φgi′(xii)φ
gj′(yj)φ
glk′(tk), (9)
2wvi,jj,k =
p(x),p(t)∑i′,k′=0
p(y)+1∑j′=−1
wi′,j′,k′φgi′(xi)φ
gj′(yjj)φ
glk′(tk), (10)
or for the non-extended element-wise flux reconstruction,
FA(wi,j , t) =−p(t)∑k=0
p(x)+1∑ii=0
uii,j,k1wuii,j,k(φglii )
′(xi)φglk (t)
−p(t)∑k=0
p(y)+1∑jj=0
vi,jj,k1wvi,jj,k(φgljj)
′(yj)φglk (t). (11)
with
1wuii,j,k =
p(x),p(y),p(t)∑i′,j′,k′=0
wi′,j′,k′φgi′(xii)φ
gj′(yj)φ
glk′(tk), (12)
1wvi,jj,k =
p(x),p(y),p(t)∑i′,j′,k′=0
wi′,j′,k′φgi′(xi)φ
gj′(yjj)φ
glk′(tk), (13)
FD(wi,j , t) is defined as follows,
FD(wi,j , t) =
p(t)∑k=0
p(x)+1∑ii=0
(D11)ii,j,k2wxii,j,k(φglii )
′(xi)φglk (t)
+
p(t)∑k=0
p(y)+1∑jj=0
(D22)i,jj,k2wyi,jj,k(φgljj)
′(yj)φglk (t)
+
p(t)∑k=0
p(x)+1∑ii=0
(D12)ii,j,k2wyii,j,k(φglii )
′(xi)φglk (t)
+
p(t)∑k=0
p(y)+1∑jj=0
(D21)i,jj,k2wxi,jj,k(φgljj)
′(yj)φglk (t), (14)
and FR(wi,j , t) is defined by
FR(wi,j , t) = fr(
p(t)∑k=0
wi,j,kφglk (t)). (15)
12 Chaoxu Pei et al.
Note that the terms 2wx and 2wy in Eq. (5.1) are computed as follows,
2wx =
p(y),p(t)∑j,k=0
p(x)+2∑ii=0
2wxii,j,k(φglii )′(x)φgj (y)φglk (t), (16)
2wy =
p(x),p(t)∑i,k=0
p(y)+2∑jj=0
2wyi,jj,kφgi (x)(φgljj)
′(y)φglk (t), (17)
with
2wxii,j,k =
p(x)+1∑i′=−1
p(y),p(t)∑j′,k′=0
wi′,j′,k′φgi′(xii)φ
gj′(yj)φ
glk′(tk), (18)
2wyi,jj,k =
p(y)+1∑j′=−1
p(x),p(t)∑i′,k′=0
wi′,j′,k′φgi′(xi)φ
gj′′(yjj)φ
glk′(tk). (19)
The points xip(x)
i=0 , yjp(y)
j=0 are sets of Gauss points; xiip(x)+1i=0 , yjjp
(y)+1j=0 ,1
and tkp(t)
k=0 are sets of Gauss-Lobatto points; xi′p(x)+1i′=−1 and yj′p
(y)+1j′=−1 are2
sets of Extended-Gauss points described in Fig. 3. Note that the flux values at3
the endpoints in each spatial element for both the hyperbolic term and viscous4
term are defined as follows.5
– The EEFR. The values at the endpoints in each spatial element, 2wu0,j,k,6
2wup(x)+1,j,k
, 2wvi,0,k, and 2wvi,p(y)+1,k
, are overwritten by either the average7
values or the upwind values at the inter-element faces.8
– The NEEFR. The values at the endpoints in each spatial element, 1wu0,j,k,9
1wup(x)+1,j,k
, 1wvi,0,k, and 1wvi,p(y)+1,k
, are overwritten by the average values10
on either side of the inter-element faces for conservation laws (see Table 1),11
and the endpoints are overwritten by either the average values or the up-12
wind values at the inter-element faces for parabolic problems.13
– Viscous flux. The values of 2wx and 2wy at the endpoints in each spatial14
element are overwritten by the average values on either side of the inter-15
element faces.16
Remark 1 Huynh [16] reported on the stability of a number of spectral ele-17
ment approaches for solving the conservation laws (see section V I in [16]), in18
which the scheme of NEEFR-upwind with the Chebyshev-Gauss points and19
the Chebyshev-Gauss-Lobatto points defined on a staggered grid is found to20
be mildly unstable. We have found that the scheme of NEEFR-upwind with21
the Legendre-Gauss points and the Legendre-Gauss-Lobatto points defined on22
a staggered grid is also mildly unstable for solving a linear advection equation23
(see section 6), which is consistent with the results reported by Huynh [16].24
We list the stability of several numerical schemes for solving conservation laws25
in Table 1.26
Multi-implicit space-time spectral element method 13
Table 1 The stability of numerical schemes for solving the conservation laws.
Numerical method Stability Nodes Extended stencilEEFR-average yes Legendre yesEEFR-upwind yes Legendre yes
NEEFR-average yes Legendre noNEEFR-upwind no Legendre/Chebyshev noDG-upwind [9] yes Legendre/Chebyshev no
Flux reconstruction [16] yes Legendre no
5.2 Multi-implicit spectral deferred correction method1
The MISDC method introduced by Bourlioux et al. [5] is a variant of the classi-2
cal spectral deferred correction (SDC) method. SDC methods are derived from3
representing an evolution equation as an integral in time, and approximating4
this integral by high order quadrature rules. To reduce the integration error,5
a series of correction equations are designed and solved by a low-order time-6
integration scheme. These correction equations can be applied iteratively to7
achieve arbitrary high-order accuracy in time. The choices of correction equa-8
tions and the efficiency of variances of the SDC methods have been discussed9
in great detail by Layton [26,27].10
A key feature of the MISDC method is that it iteratively couples all phys-ical processes together by including the effects of each process during the in-tegration of any particular process. In contrast, traditional operator-splittingmethods ignore the effects of other processes, that is, each process is discretizedin isolation. After the spatial discretization of the advection-diffusion-reactionequation, the resulting ODE system (7) is rewritten as
∂w
∂t= FA(w) + FD(w) + FR(w) = F (w(t), t), (20)
where FA(w) denotes the spatial discretization of the advection term ∇·(uw),FD denotes the spatial discretization of the diffusion term ∇ · (D∇w) and FRdenotes the spatial discretization of the reaction term fr(w). Since the advec-tion, diffusion and reaction processes are on distinct time scales, one would liketo use an explicit treatment for advection while an implicit treatment for bothdiffusion and reaction in order to avoid a stringent time step constraint. Acomplicated system of different processes can be solved in a decoupled fashionby applying an operator splitting, but the overall order of accuracy is restrictedby the splitting error. For example, the solution advanced in one time step by
14 Chaoxu Pei et al.
an operator splitting method can be obtained as follows,
wA(tn+1) = w(tn) +
∫ tn+1
tnFA(wA(τ))dτ, (21)
wD(tn+1) = wA(tn+1) +
∫ tn+1
tnFD(wD(τ))dτ, (22)
w(tn+1) = wD(tn+1) +
∫ tn+1
tnFR(w(τ))dτ. (23)
One can show that the above approximation is O(∆t) globally unlessthe operators associated with FA, FD and FR commute. In comparison, theMISDC method can achieve higher order of accuracy by the iterative cou-pling strategy which reduces both the splitting error and the integration er-ror. Pazner et al. [32] proposed an iterative coupling strategy in the deferredcorrection procedure as follows,
wn+1,k+1A = wn,k+1 +
∫ tn+1
tn[FA(wk+1
A )− FA(wk)]dτ
+
∫ tn+1
tn[FA(wk) + FD(wk) + FR(wk)]dτ, (24)
wn+1,k+1AD = wn,k+1 +
∫ tn+1
tn[FA(wk+1
A )− FA(wk)]dτ
+
∫ tn+1
tn[FA(wk) + FD(wk) + FR(wk)]dτ
+
∫ tn+1
tn[FD(wk+1
AD )− FD(wk)]dτ, (25)
wn+1,k+1 = wn,k+1 +
∫ tn+1
tn[FA(wk+1
A )− FA(wk)]dτ
+
∫ tn+1
tn[FA(wk) + FD(wk) + FR(wk)]dτ
+
∫ tn+1
tn[FD(wk+1
AD )− FD(wk)]dτ
+
∫ tn+1
tn[FR(wk+1)− FR(wk)]dτ. (26)
Overall fourth order of accuracy has been demonstrated by Pazner et al. [32] on1
one-dimensional low Mach number flow. They pointed out that either reducing2
the time step size or increasing the number of iterations per step can achieve3
the fourth-order accuracy. For example, 8 iterations per step was sufficient to4
achieve the fourth-order accuracy in all tests presented in [32].5
Multi-implicit space-time spectral element method 15
5.2.1 Two different coupling strategies1
Based on the idea of iterative coupling, we propose a new multi-implicit SDCcoupling scheme as follows,
wn+1,k+1A = wn,k+1 +
∫ tn+1
tn[FA(wk+1
A )− FA(wkA)]dτ +
∫ tn+1
tn[FA(wk)]dτ,
(27a)
wn+1,k+1AD = wn+1,k+1
A +
∫ tn+1
tn[FD(wk+1
AD )− FD(wkAD)]dτ +
∫ tn+1
tn[FD(wk)]dτ,
(27b)
wn+1,k+1 = wn+1,k+1AD +
∫ tn+1
tn[FR(wk+1)− FR(wk)]dτ +
∫ tn+1
tn[FR(wk)]dτ.
(27c)
In order to construct an arbitrary high-order time integration scheme, thesecond integrals in (27) are evaluated by a higher-order quadrature rule, e.g.,the Gauss quadrature. Let Im+1
m (F (wk)) be a numerical quadrature, e.g., theGauss quadrature, an approximation to∫ tnm+1
tnm
F (wk(τ), τ)dτ. (28)
On the other hand, the first integrals are discretized by a low-order time2
integration scheme, e.g., a first order time integration scheme.3
One might notice that both flux reconstructions, including informationfrom all neighbors, are analogous to high order centered difference schemes [3].Along with the spatial discretization of high-order center differences, Bao andJin [3] applied a fourth order I-stable method in time that allows for a verylarge cell Reynolds number when solving a system of convection-diffusion equa-tions with a small viscosity. The region of absolute stability of their I-stablemethod includes part of the imaginary axis. Motivated by such a temporalapproach, we introduce a first-order I-stable building block time integrationscheme along with either the EEFR or the NEEFR as an explicit treatmentfor the advection process.
w∗i,j = (wi,j)m +∆tmFA((wi,j)m, tnm), (29a)
(wi,j)m+1 = (wi,j)m +∆tmFA((wi,j)∗, tnm+1). (29b)
The region of absolute stability (or simply the stability region) of an I-stable4
method contains part of the imaginary axis. In Fig.6, the stability regions of5
both I-stable scheme (black) and forward Euler (red) are illustrated.6
The discretization of the present multi-implicit space-time spectral element7
method with new coupling strategy is described in Algorithm 1 (indices i and8
j in Eq. (7) are suppressed for simplicity), in which the extended element-wise9
16 Chaoxu Pei et al.
Fig. 6 An illustration of the stability regions of I-stable scheme (black) and forward Euler(red). Both time integration schemes are first order.
flux reconstruction is employed for treating the advection term. The coupling1
strategy applied in Algorithm 1 is referred to as coupling 1.2
In addition, a second coupling strategy, by extending the method of Pazneret al. [32], is introduced as follows
wn+1,k+1A = wn,k+1 +
∫ tn+1
tn[FA(wk+1
A )− FA(wkA)]dτ +
∫ tn+1
tn[FA(wk)]dτ,
(30a)
wn+1,k+1ADR = wn+1,k+1
A +
∫ tn+1
tn[FD(wk+1
ADR)− FD(wk)]dτ
+
∫ tn+1
tn[FD(wk) + FR(wk)]dτ, (30b)
wn+1,k+1 = wn+1,k+1ADR +
∫ tn+1
tn[FR(wk+1)− FR(wk)]dτ. (30c)
The procedure of such a coupling strategy is described in the Algorithm 23
(indices i and j in Eq. (7) are suppressed for simplicity), which is referred to4
as coupling 2. Comparing Algorithm 1 to Algorithm 2, we see that steps 175
and 20 in Algorithm 2 are different from those in Algorithm 1 due to different6
coupling strategies employed.7
With a coupling strategy (coupling 1 or coupling 2) used in a series ofdeferred correction steps, the temporal splitting error is eliminated iteratively.Thus, we conjecture that our multi-implicit space-time spectral element meth-ods, Algorithm 1 and Algorithm 2, have an overall order of accuracy
minp(x) + 1, p(y) + 1, p(t) + 1,K, (31)
Multi-implicit space-time spectral element method 17
Algorithm 1 The multi-implicit space-time spectral element method usingthe extended element-wise reconstruction with I-stable building block scheme– Coupling 1.
1: for m = 0, .., p(t) − 1 do2: FA(w0
m, tnm) = 0.
3: FA(w0,∗, tnm) = 0.4: Im+1
m (FA(w0)) = 0.5: FD((wAD)0m+1, t
nm+1) = 0.
6: Im+1m (FD(w0)) = 0.
7: FR(w0m+1, t
nm+1) = 0.
8: Im+1m (FR(w0)) = 0.
9: end for10: for k = 1, ..,K do11: wk
0 = w(tn).
12: for m = 0, .., p(t) − 1 do13: Compute FA(wk
m, tnm) by Eq. (8). . Advection process (13-16)
14: (wA)k,∗ = wkm +∆tm[FA(wk
m, tnm)− FA(wk−1
m , tnm)] + Im+1m (FA(wk−1)).
15: Compute FA(wk,∗, tnm) by Eq. (8).16: (wA)km+1 = wk
m +∆tm[FA((wA)k,∗, tnm+1)− FA((wA)k−1,∗, tnm+1)]
+Im+1m (FA(wk−1)).
17: (wAD)km+1 = (wA)km+1 +∆tm[FD((wAD)km+1, tnm+1)−
FD((wAD)k−1m+1, t
nm+1)] + Im+1
m (FD(wk−1)).
18: (wAD)km+1 is computed by an iterative method.
19: Compute FD((wAD)km+1, tnm+1) by Eq. (14). . Diffusion process (17-19)
20: wkm+1 = (wAD)km+1 +∆tm[FR(wk
m+1, tnm+1)− FR(wk−1
m+1, tnm+1)]
+Im+1m (FR(wk−1)).
21: wkm+1 is computed by Newton’s method.
22: Compute FR(wkm+1, t
nm+1) by Eq. (15). . Reaction process (20-22)
23: end for24: Compute Im+1
m (FA(wk)), Im+1m (FD(wk)), and Im+1
m (FR(wk)) with the updatedsolutions wk.
25: end for
where K is the number of iterations in the multi-implicit SDC method, p(x),1
p(y) and p(t) is the polynomial order in the x-direction, the y-direction and2
the temporal direction, respectively.3
5.3 Efficiency of the low order building block in the deferred correction4
procedure5
The first-order I-stable building block time integration scheme in Eq. (29) isa two-stage explicit Runge-Kutta method for integrating the advection term.In section 5.2.1, each stage of the first-order I-stable building block time inte-gration scheme is corrected in the deferred correction phase of the space-timealgorithm, and the discretization is illustrated in steps 14 and 16 in both Al-gorithm 1 and Algorithm 2. This correction strategy in Eq. (32) is referredto as the “intrusive correction” because the predictor time integration scheme
18 Chaoxu Pei et al.
Algorithm 2 The multi-implicit space-time spectral element method usingthe extended element-wise reconstruction with I-stable building block schemeand the coupling strategy obtained by extending the method of Pazner etal. [32] – Coupling 2.
1: for m = 0, .., p(t) − 1 do2: FA(w0
m, tnm) = 0.
3: FA(w0,∗, tnm) = 0.4: Im+1
m (FA(w0)) = 0.5: FD((wADR)0m+1, t
nm+1) = 0.
6: Im+1m (FD(w0)) = 0.
7: FR(w0m+1, t
nm+1) = 0.
8: Im+1m (FR(w0)) = 0.
9: end for10: for k = 1, ..,K do11: wk
0 = w(tn).
12: for m = 0, .., p(t) − 1 do13: Compute FA(wk
m, tnm) by Eq. (8). . Advection process (13-16)
14: (wA)k,∗ = wkm +∆tm[FA(wk
m, tnm)− FA(wk−1
m , tnm)] + Im+1m (FA(wk−1)).
15: Compute FA(wk,∗, tnm) by Eq. (8).16: (wA)km+1 = wk
m +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)]
+Im+1m (FA(wk−1)).
17: (wADR)km+1 = (wA)km+1 +∆tm[FD((wADR)km+1, tnm+1)−
FD(wk−1m+1, t
nm+1)] + Im+1
m (FD(wk−1)) + Im+1m (FR(wk−1)).
18: (wADR)km+1 is computed by an iterative method.
19: Compute FD((wADR)km+1, tnm+1) by Eq. (14). . Diffusion process (17-19)
20: wkm+1 = (wADR)km+1 +∆tm[FR(wk
m+1, tnm+1)− FR(wk−1
m+1, tnm+1)].
21: wkm+1 is computed by Newton’s method.
22: Compute FR(wkm+1, t
nm+1) by Eq. (15). . Reaction process (20-22)
23: end for24: Compute Im+1
m (FA(wk)), Im+1m (FD(wk)), and Im+1
m (FR(wk)) with the updatedsolutions wk.
25: end for
cannot be treated as a “black box.”
(wA)k,∗ = wkm +∆tm[FA(wkm, tnm)− FA(wk−1m , tnm)] + Im+1
m (FA(wk−1)),(32a)
(wA)km+1 = wkm +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)] + Im+1m (FA(wk−1)).
(32b)
We have found that an alternate, “non-intrusive,” correction algorithm isjust as accurate as the “intrusive” algorithm and is more memory and cpuefficient. The “non-intrusive” approach simply ignores the correction of thefirst stage in the two stage I-scheme. Such a correction strategy in Eq. (33)is referred to as “non-intrusive correction”. We analyze the “non-intrusive”
Multi-implicit space-time spectral element method 19
method in section 6.
(wA)k,∗ = wkm +∆tm[FA(wkm, tnm)], (33a)
(wA)km+1 = wkm +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)] + Im+1m (FA(wk−1)).
(33b)
6 Numerical tests1
In this section, we test the multi-implicit space-time spectral element methodfor solving problems with multiple time scales in two spatial dimensions. Thenumerical results are compared with the exact solutions in order to examinethe spectral accuracy in both space and time. An explicit time integrationscheme is applied on the advection process, which leads to the following CFLcondition for the time-substep size
max∆tm = C(min∆xn|u|max
+min∆yn|v|max
), (34)
where ∆tm is the time substep size (max∆tm ≤ ∆t), C is a constant that is2
less than 1.0, and min∆xn (or min∆yn) is the smallest distance between3
two Gauss points that is proportional to the inverse of the square of polynomial4
order, i.e., 1/(p(x))2 (or 1/(p(y))2) in the x-direction (or y-direction) [22]. The5
discrete system obtained from the diffusion process is well conditioned and6
can be solved in a very efficient way via the biconjugate gradient stabilized7
method (BiCGSTAB), even without the use of any preconditioner.8
In the following tests, we keep the number of iterations K in the multi-implicit SDC method as K = p(t) unless the value of K is prescribed. In the
last space-time slab EE(t)−1, the errors are measured in the discrete L∞ norms
at time tE(t)
= T
‖Errw‖∞ = maxi=0,...,p(x),
j=0,...,p(y)
‖wE(t)
i,j − (wh)E(t)
i,j ‖∞, (35)
where T is the finial computational time, wE(t)
i,j denotes the exact solution eval-9
uated at the node (xi, yj), and (wh)E(t)
i,j is the approximate solution evaluated10
at the same node.11
6.0.1 Treatments for the advection process12
We consider the following advection problem in a Cartesian domain Ω =[0, 1]× [0, 1]
wt +∇ · (uw) = 0, x ∈ Ω, (36)
with the exact solution
w(x, t) = sin(2π(x− ut)) sin(2π(y − vt)). (37)
20 Chaoxu Pei et al.
Periodic boundary conditions are imposed, and the velocity vector u = (u, v)>1
is set to be (1, 1)>.2
First, we demonstrate the spectral accuracy of our method by plotting the3
maximum errors of the solution as a function of polynomial orders, (p(x), p(y), p(t)) =4
(p, p, p(t)). The computational domain is divided into 4 × 4 and 8 × 8 spatial5
elements, and the final time of the computation is T = 0.5. In Fig. 7, we6
see that both the extended element-wise flux reconstruction (EEFR) and the7
non-extended element-wise flux reconstruction (NEEFR) exhibit spectral ac-8
curacy in space when the number of space-time slab is fixed, i.e., E(t) = 160.9
The errors of all three schemes decrease as the resolution in space increases.10
The results of the NEEFR with the upwind values are not shown because11
the scheme is mildly unstable for solving conservation laws (see Remark 1),12
which will be discussed in detail later. In addition, we compare two correction
1 2 3 4 5 6 7 8 91011
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p;EEFR−average;
4 × 4
8 × 8
1 2 3 4 5 6 7 8 9101110
−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p;EEFR−upwind;
4 × 4
8 × 8
1 2 3 4 5 6 7 8 9101110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p;NEEFR−average;
4 × 4
8 × 8
Fig. 7 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with E(t) = 160. The polynomial order in time p(t) is chosen to thesame as the one in space p. The left one is the results obtained by the EEFR-average withtwo different spatial tessellations (4 × 4 and 8 × 8), the middle one is the results obtainedby the EEFR-upwind with two different spatial tessellations, and the right one is the resultsobtained by the NEEFR-average with two different spatial tessellations.
13
strategies, “intrusive correction” and “non-intrusive correction”. In Fig. 8, we14
see that both options have the same error behavior.15
We then compare the performance of all three schemes: the EEFR with the16
average values (EEFR-average), the EEFR with the upwind values (EEFR-17
upwind), and the NEEFR with the average values (NEEFR-average). The18
comparison results are shown in Fig. 9. The schemes with the EEFR (EEFR-19
average and EEFR-upwind) are more accurate than the scheme with the20
Multi-implicit space-time spectral element method 21
1 2 3 4 5 6 7 8 9 101110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; EEFR−average;
Intrusive
Non−intrusive
1 2 3 4 5 6 7 8 9 101110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; EEFR−upwind;
Intrusive
Non−intrusive
1 2 3 4 5 6 7 8 9 101110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; NEEFR−average;
Intrusive
Non−intrusive
Fig. 8 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulationis computed up to 0.5 with (E(x) × E(y), E(t)) = (4 × 4, 160). The polynomial order intime p(t) is chosen to the same as the one in space p. The left one is the results obtainedby the EEFR-average with two different correction strategies (“intrusive correction” and“non-intrusive correction”), the middle one is the results obtained by the EEFR-upwindwith two different correction strategies, and the right one is the results obtained by theNEEFR-average with two different correction strategies.
NEEFR (NEEFR-average) when using the same number of degrees of freedom.1
Thus, the schemes with the EEFR as the treatment for the advection process2
are more accurate and efficient than those with the NEEFR. Furthermore,3
there is no much difference in the error behavior between the EEFR-average4
and the EEFR-upwind.5
Next, we examine the performance of our multi-implicit space-time method6
with respect to the number of space-time slab E(t). Fixing the mesh in space7
to be 8 × 8, we test the model equation (36) with E(t) = 80 and E(t) = 160.8
Note that the minimum value of the number of space-time slab is limited9
by the CFL condition because the explicit I-stable time integration scheme10
is employed. In Fig. 10, we see that all three schemes (the EEFR-average,11
the EEFR-upwind and the NEEFR-average) exhibit spectral accuracy in time12
with the polynomial order in space equal to the one in time: p = p(t). In13
addition, the comparison among all three schemes is shown in Fig. 11. In14
addition, two correction strategies, “intrusive correction” and “non-intrusive15
correction”, are compared. The results are shown in Fig. 12. It is clear that16
both correction strategies have the same error behavior.17
Now, we report results which indicate that the NEEFR scheme using the18
upwind values at the endpoints (NEEFR-upwind) in each spatial element is19
22 Chaoxu Pei et al.
2 3 4 5 6 7 8 9 10 1110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 4 × 4;
EEFR−average
EEFR−upwind
NEEFR−average
2 3 4 5 6 7 8 9 10 1110
−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 8 × 8;
EEFR−average
EEFR−upwind
NEEFR−average
Fig. 9 Errors in the solution as a function of polynomial order in space p for two differentspatial tessellations: 4× 4 and 8× 8. The simulation is computed up to 0.5 with E(t) = 160.The polynomial order in time p(t) is chosen to the same as the one in space p. The left oneis the comparison among the EEFR-average, the EEFR-upwind and the NEEFR-averagewith 4×4 spatial elements, while the right one is the comparison among these three schemeswith 8× 8 spatial elements.
1 2 3 4 5 6 7 8 91011
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
;EEFR−average;
E(t)
=80
E(t)
=160
1 2 3 4 5 6 7 8 91011
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
;EEFR−upwind;
E(t)
=80
E(t)
=160
1 2 3 4 5 6 7 8 9101110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
;NEEFR−average;
E(t)
=80
E(t)
=160
Fig. 10 Errors in the solution as a function of polynomial order in time p(t) for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with 8× 8 numbers of spatial elements. The pair of polynomial ordersin space, (p(x), p(y)) = (p, p), is chosen to the same as the one in time p(t). The left oneis the results obtained by the EEFR-average with two different numbers of space-time slab(E(t) = 80 and E(t) = 160), the middle one is the results obtained by the EEFR-upwindwith two different numbers of space-time slab, and the right one is the results obtained bythe NEEFR-average with two different numbers of space-time slab.
Multi-implicit space-time spectral element method 23
2 3 4 5 6 7 8 9 10 1110
−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; E(t)
=80;
EEFR−average
EEFR−upwind
NEEFR−average
2 3 4 5 6 7 8 9 10 1110
−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; E(t)
=160;
EEFR−average
EEFR−upwind
NEEFR−average
Fig. 11 Errors in the solution as a function of polynomial order in time p(t) for two differentnumbers of space-time slab: E(t) = 80 and E(t) = 160. The simulation is computed up to 0.5with 8× 8 spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p),is chosen to the same as the one in time p(t). The left one is the comparison among theEEFR-average, the EEFR-upwind and the NEEFR-average with E(t) = 80, while the rightone is the comparison among these three schemes with E(t) = 160.
mildly unstable for solving conservation laws. We compare the EEFR-average,1
the EEFR-upwind, the NEEFR-average and the NEEFR-upwind with 8 × 82
spatial tessellation for solving the linear advection equation (36). The compu-3
tation is carried out up to T = 1.0, and the results are shown in Fig. 13. On the4
left panel, four schemes are compared when E(t) = 200. The stable schemes,5
the EEFR-average, the EEFR-upwind and the NEEFR-average, exhibit the6
spectral accuracy: the error decays exponentially fast until reaching 10−147
and staying around there as the polynomial order increases. By contrast, the8
mildly unstable scheme, the NEEFR-upwind, has an error behavior of first9
exponentially decaying until reaching 10−11 at polynomial order p = 9 and10
increasing afterwords. In order to exclude the possibility that the CFL con-11
straint for the NEEFR-upwind is smaller, we also test it with more space-time12
slabs, i.e., E(t) = 500. The result is shown on the right panel of the Fig. 13.13
The error in both cases exhibit the same behavior, that is, it increases after14
reaching 10−11. In addition, two correction strategies, “intrusive correction”15
and “non-intrusive correction”, are compared. The results of four schemes are16
shown in Fig. 14. It is clear that both correction strategies have the same error17
behavior.18
24 Chaoxu Pei et al.
1 2 3 4 5 6 7 8 9 1011
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; EEFR−average;
Intrusive
Non−intrusive
1 2 3 4 5 6 7 8 9 1011
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; EEFR−upwind;
Intrusive
Non−intrusive
1 2 3 4 5 6 7 8 9 101110
−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; NEEFR−average;
Intrusive
Non−intrusive
Fig. 12 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with (E(x)×E(y), E(t)) = (8×8, 80). The polynomial order in time p(t)
is chosen to the same as the one in space p. The left one is the results obtained by the EEFR-average with two different correction strategies, the middle one is the results obtained bythe EEFR-upwind with two different correction strategies (“intrusive correction” and “non-intrusive correction”), and the right one is the results obtained by the NEEFR-average withtwo different correction strategies.
6.0.2 Advection-diffusion-reaction problems1
On a computational domain Ω = [0, 1] × [0, 1], we consider the followingadvection-diffusion-reaction problem
wt +∇ · (uw) = ν∆w + λw, (38)
with an exact solution
w(x, t) = 3e−(2(2π)2ν−λ)t sin(2π(x− ut)) sin(2π(y − vt)). (39)
Periodic boundary conditions are imposed, and the velocity vector u = (u, v)>2
is set to be (1, 1)>. We also choose ν = 0.04 and λ = 5.0 in the following tests.3
The performance of the two advective flux reconstructions is compared4
with each coupling strategies, coupling 1 (Algorithm 1) and coupling 2 (Al-5
gorithm 2), on the space-time tessellation, (E(x) × E(y), E(t)) = (5 × 5, 80).6
The final time of the computation is T = 1.0, and the results are shown in7
Fig. 15. The left panel is the comparison made among the EEFR-average, the8
EEFR-upwind, the NEEFR-average and the NEEFR-upwind with coupling9
1 (Algorithm 1), and the one on the right is the comparison made among10
Multi-implicit space-time spectral element method 25
2 3 4 5 6 7 8 9 10 11
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p;10 × 10;E(t)
=200;
EEFR−average
EEFR−upwind
NEEFR−average
NEEFR−upwind
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p;10 × 10;NEEFR−upwind;
E(t)
=200
E(t)
=500
Fig. 13 Errors in the solution of the advection equation (36) as a function of polynomialorder in space p. The simulation is computed up to 1.0 with 8 × 8 spatial tessellation.The polynomial order in time, p(t), is chosen to the same as the one in space p. The leftone is the comparison among the EEFR-average, the EEFR-upwind, the NEEFR-averageand the NEEFR-upwind when using E(t) = 200. The right one is the comparison of theNEEFR-upwind between different numbers of space-time slab: E(t) = 200 and E(t) = 500.
the EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-1
upwind with coupling 2 (Algorithm 2). Both coupling strategies with each2
treatment for the advective process exhibit spectral accuracy. Since the tol-3
erance is set to be 10−11 in the correction step, the error decreases until it4
reaches 10−10. Clearly, the schemes with the EEFR are more accurate and5
efficient than those with the NEEFR. In the current test, coupling strategies6
with the EEFR-upwind are slightly more accurate than those with the EEFR-7
average, even though their error behavior is indeed similar. In addition, two8
correction strategies, “intrusive correction” and “non-intrusive correction”, are9
compared. The results of four schemes with two different coupling strategies10
are shown in Fig. 16, in which it is shown that both correction strategies have11
the same error behavior.12
Next, we look closely into the performance of schemes with two different13
coupling strategies using the EEFR-upwind as the treatment for the advection14
process. We demonstrate the spectral accuracy of our method by plotting15
the maximum error as a function of the polynomial orders (p(x), p(y), p(t)) =16
(p, p, p(t)). The computational domain is divided into 5×5 and 10×10 spatial17
elements, respectively. The final time is T = 1.0. In Fig. 17, we see that both18
coupling strategies (Algorithm 1 and Algorithm 2) exhibit spectral accuracy in19
space when the number of space-time slab is fixed, i.e., E(t) = 160. The errors20
of both schemes decrease as the resolution in space increases. We also compare21
the performance of the two different coupling strategies, and the comparison22
is shown in Fig. 9. Two different spatial tessellations, 5 × 5 and 10 × 10, are23
26 Chaoxu Pei et al.
2 3 4 5 6 7 8 9 10 11
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 10 × 10; E(t)
=200; EEFR−average
Intrusive
Non−intrusive
2 3 4 5 6 7 8 9 10 11
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 10 × 10; E(t)
=200; EEFR−upwind
Intrusive
Non−intrusive
2 3 4 5 6 7 8 9 10 11
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 10 × 10; E(t)
=200; NEEFR−average
Intrusive
Non−intrusive
2 3 4 5 6 7 8 9 10 11
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 10 × 10; E(t)
=200; NEEFR−upwind
Fully correction
Semi−correction
Fig. 14 Errors in the solution of the advection equation (36) as a function of polynomialorder in space p. The simulation is computed up to 1.0 with (E(x)×E(y), E(t)) = (8×8, 200).The polynomial order in time, p(t), is chosen to the same as the one in space p. On the toppanel, it is the comparison between two correction strategies (“intrusive correction” and“non-intrusive correction”) along with either the EEFR-average (left) or the EEFR-upwind(right). On the bottom panel, it is the comparison among two correction strategies alongwith either the NEEFR-average (left) or the NEEFR-upwind (right).
tested. We find that there is no much difference in the error behavior of the1
two different coupling strategies in the current test case.2
We then examine the performance of our multi-implicit space-time method3
with respect to the number of space-time slab E(t). With the 10× 10 spatial4
mesh, we test our method with E(t) = 80 and E(t) = 160, respectively. Note5
again that the minimum value of the number of space-time slab is dictated6
by the CFL condition due to the explicit treatment of the advection process.7
In Fig. 19, we see that both coupling strategies exhibit spectral accuracy in8
time with the polynomial order in space equal to the one in time: p = p(t).9
In addition, the comparison between the two different coupling strategies is10
shown in Fig. 20.11
In Tables 2, 3 and 4, we report on the performance of our two differ-ent space-time coupling strategies (Algorithm 1 and Algorithm 2), using theEEFR-upwind as the treatment for the advection process, by varying the spa-tial tessellation E(x)×E(y), the number of space-time slab E(t) and the numberof iterations K. The order of convergence, r, listed in tables are found by
r = log2(‖Errh‖∞‖Errh/2‖∞
). (40)
Multi-implicit space-time spectral element method 27
2 3 4 5 6 7 8 9 10 1110
−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Poly. Order p
||E
RR
||m
ax
p(t)
=p; Coupling 1;
EEFR−average
EEFR−upwind
NEEFR−average
NEEFR−upwind
2 3 4 5 6 7 8 9 10 1110
−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Poly. Order p
||E
RR
||m
ax
p(t)
=p; Coupling 2;
EEFR−average
EEFR−upwind
NEEFR−average
NEEFR−upwind
Fig. 15 Errors in the solution as a function of polynomial order in space p. The simulationis computed up to 1.0 with 5×5 spatial tessellation and E(t) = 80. The polynomial order intime, p(t), is chosen to the same as the one in space p. The left part is the comparison madeamong the EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-upwindwith coupling 1 (Algorithm 1), while the one on the right is the comparison made amongthe EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-upwind withcoupling 2 (Algorithm 2). The tolerance is set to be 10−11.
The results reported in Table 2 are in agreement with the analytic formula (31),1
in which K-th order of accuracy is obtained. By comparing the results in Table2
3 with Table 4, we see that either reducing the time step (i.e., increasing the3
number of space-time slab E(t)) or increasing the number of iterations per4
step (i.e., increasing K) can achieve the desired order of accuracy: p(t) + 1. By5
comparing the errors in Tables 3 and 4, the strategy of increasing the number6
of iterations per time step is more accurate than the strategy of reducing the7
time step.8
Now, we investigate the convergence rate in space and in time, separately.9
In order to examine the order of convergence in space, the polynomial order10
in time is set to be higher than the one in space, so that the temporal error11
is negligible. In Table 5, the temporal polynomial order and the number of12
iterations are set to be p(t) = 6 and K = 8. We test different polynomial13
orders in space from p = 4 to p = 7. The results in Table 5 show that p + 114
order of convergence in space is obtained.15
To test the order of convergence in time, the polynomial order in space is16
set to be higher than the one in time, so that the spatial error is negligible.17
The polynomial order in space is chosen to be either p = 8 or p = 9, and then18
combinations of temporal order and the number of iterations are varied. The19
results are shown in Table 6, and the global order of accuracy is observed to20
be minp(t) + 1,K.21
28 Chaoxu Pei et al.
2 3 4 5 6 7 8 9 10 11
10−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100
Poly. Order p
||ER
R|| m
ax
p(t)=p; EEFR−average; Coupling 1;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; EEFR−average; Coupling 2;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; EEFR−upwind; Coupling 1;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; EEFR−upwind; Coupling 2;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 11
10−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100
Poly. Order p
||ER
R|| m
ax
p(t)=p; NEEFR−average; Coupling 1;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; NEEFR−average; Coupling 2;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; NEEFR−upwind; Coupling 1;
IntrusiveNon−intrusive
2 3 4 5 6 7 8 9 10 1110
−1010
−910
−810
−710
−610
−510
−410
−310
−210
−110
0
Poly. Order p
||ER
R|| m
ax
p(t)=p; NEEFR−upwind; Coupling 2;
IntrusiveNon−intrusive
Fig. 16 Errors in the solution as a function of polynomial order in space p. The simulationis computed up to 1.0 with (E(x) ×E(y), E(t)) = (5× 5, 80). The polynomial order in time,p(t), is chosen to the same as the one in space p. The left part is the comparison madebetween two correction strategies (“intrusive correction” and “non-intrusive correction”)along with coupling 1 (Algorithm 1) and one of the four schemes: the EEFR-average, theEEFR-upwind, the NEEFR-average and the NEEFR-upwind; the one on the right is thecomparison made between two correction strategies along with coupling 1 (Algorithm 2)and one of the four schemes: the EEFR-average, the EEFR-upwind, the NEEFR-averageand the NEEFR-upwind. The tolerance is set to be 10−11.
Multi-implicit space-time spectral element method 29
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; Coupling 1;
5 × 5
10 × 10
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; Coupling 2;
5 × 5
10 × 10
Fig. 17 Errors in the solution as a function of polynomial order in space p for two differentcoupling strategies, Algorithm 1 and Algorithm 2, using the EEFR-upwind as the treatmentfor the advection process. The simulation is computed up to 1.0 and E(t) = 160. Thepolynomial order in time p(t) is chosen to the same as the one in space p. The left one is theresults obtained by coupling 1 (Algorithm 1) with two different spatial tessellations (5 × 5and 10 × 10), while the right one is the results obtained by coupling 2 (Algorithm 2) withtwo different spatial tessellations (5× 5 and 10× 10). The tolerance is set to be 10−13.
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 5 × 5;
Coupling 1
Coupling 2
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p
||E
RR
||m
ax
p(t)
=p; 10 × 10;
Coupling 1
Coupling 2
Fig. 18 Errors in the solution as a function of polynomial order in space p for two differentspatial tessellations: 5×5 and 10×10. The simulation is computed up to 1.0 and E(t) = 160.The polynomial order in time p(t) is chosen to the same as the one in space p. The left one isthe comparison between coupling 1 with the EEFR-upwind and coupling 2 with the EEFR-upwind on the 5× 5 spatial tessellation, while the right one is the comparison between twodifferent coupling strategies on 10×10 spatial tessellation. The tolerance is set to be 10−13.
30 Chaoxu Pei et al.
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; Coupling 1;
E(t)
=80
E(t)
=160
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; Coupling 2;
E(t)
=80
E(t)
=160
Fig. 19 Errors in the solution as a function of polynomial order in time p(t) for two dif-ferent coupling strategies: coupling 1 (Algorithm 1) with the EEFR-upwind and coupling 2(Algorithm 2) with the EEFR-upwind. The simulation is computed up to 1.0 with 10× 10spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p), is chosen tothe same as the one in time p(t). The left one is the results obtained by coupling 1 using theEEFR-upwind with two different numbers of space-time slab (E(t) = 80 and E(t) = 160),while the right one is the results obtained by coupling 2 using the EEFR-upwind with twodifferent numbers of space-time slab. The tolerance is set to be 10−13.
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; E(t)
=80;
Coupling 1
Coupling 2
2 3 4 5 6 7 8 9 10 1110
−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Poly. Order p(t)
||E
RR
||m
ax
p=p(t)
; E(t)
=160;
Coupling 1
Coupling 2
Fig. 20 Errors in the solution as a function of polynomial order in time p(t) for two differentnumbers of space-time slab: E(t) = 80 and E(t) = 160. The simulation is computed up to 1.0with 10×10 spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p),is chosen to the same as the one in time p(t). The left one is the comparison between twodifferent coupling strategies with E(t) = 80, while the right one is the comparison twodifferent coupling strategies with E(t) = 160. The tolerance is set to be 10−13.
Multi-implicit space-time spectral element method 31
Table 2 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t). Thetolerance is set to be 10−13, and the final time is T = 1.0.
Overall order of convergence
(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) Coupling 1 Coupling 2
(2, 2), (2, 2)(5× 5, 40) — —
(10× 10, 80) 2.1 2.1(20× 20, 160) 2.0 2.0
(3, 3), (3, 3)(5× 5, 40) — —
(10× 10, 80) 3.0 3.0(20× 20, 160) 3.0 3.0
(4, 4), (4, 4)(5× 5, 40) — —
(10× 10, 80) 4.1 4.1(20× 20, 160) 4.0 4.0
(5, 5), (5, 5)(5× 5, 40) — —
(10× 10, 80) 5.0 5.1(20× 20, 160) 5.0 5.0
(6, 6), (6, 6)(5× 5, 40) — —
(10× 10, 80) 6.2 6.2(20× 20, 160) 5.9 5.9
Table 3 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t). Thetolerance is set to be 10−13, and the final time is T = 1.0.
Overall order of convergence
(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) ‖Err‖∞ (Coupling 1) Coupling 1 Coupling 2
(2, 2), (2, 2)(5× 5, 80) 0.917 — —
(10× 10, 160) 0.151 2.6 2.6(20× 20, 320) 3.529E-002 2.0 2.1
(3, 3), (3, 3)(5× 5, 80) 8.738E-003 — —
(10× 10, 160) 5.349E-004 4.0 4.0(20× 20, 320) 5.938E-005 3.1 3.1
(4, 4), (4, 4)(5× 5, 80) 6.460E-004 — —
(10× 10, 160) 1.822E-005 5.1 5.2(20× 20, 320) 8.479E-007 4.4 4.4
(5, 5), (5, 5)(5× 5, 80) 1.763E-005 — —
(10× 10, 160) 3.372E-007 5.7 5.7(20× 20, 320) 6.874E-009 5.6 5.6
(6, 6), (6, 6)(5× 5, 80) 5.480E-007 — —
(10× 10, 160) 1.626E-009 8.3 8.3(20× 20, 320) 3.794E-011 5.4 5.4
7 Conclusions1
Novel multi-implicit space-time spectral element methods are proposed to solve2
problems associated with two or more processes with differing characteristic3
time scales. Different from the fully implicit space-time method [11,33], the4
method of lines approach is employed in order to avoid the effort of solving5
a large nonlinear system of equations constituted by the degrees of freedom6
in both space and time. The discontinuous spectral element method in the7
32 Chaoxu Pei et al.
Table 4 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t) + 1.The tolerance is set to be 10−13, and the final time is T = 1.0.
Overall order of convergence
(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) ‖Err‖∞ (Coupling 1) Coupling 1 Coupling 2
(2, 2), (2, 3)(5× 5, 40) 0.316 — —
(10× 10, 80) 1.893E-002 4.0 4.1(20× 20, 160) 1.747E-003 3.4 3.4
(3, 3), (3, 4)(5× 5, 40) 2.002E-002 — —
(10× 10, 80) 1.158E-003 4.1 4.1(20× 20, 160) 7.643E-005 3.9 3.9
(4, 4), (4, 5)(5× 5, 40) 3.250E-004 — —
(10× 10, 80) 7.788E-006 5.3 5.7(20× 20, 160) 2.316E-007 5.0 5.1
(5, 5), (5, 6)(5× 5, 40) 1.520E-005 — —
(10× 10, 80) 2.313E-007 6.0 6.0(20× 20, 160) 3.923E-009 5.8 5.8
(6, 6), (6, 7)(5× 5, 40) 6.832E-007 — —
(10× 10, 80) 2.693E-009 7.9 8.0(20× 20, 160) 5.261E-011 5.6 5.5
Table 5 Order of convergence in space of the present multi-implicit space-time spectralelement method for two different coupling strategies: coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind. The tolerance is setto be 10−13, and the final time is T = 1.0.
Order of convergence in space
(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) Coupling 1 Coupling 2
(4, 4), (6, 8)(3× 3, 100) — —(6× 6, 100) 5.7 5.7
(12× 12, 100) 6.0 6.0
(5,5) ,(6,8)(3× 3, 100) — —(6× 6, 100) 6.5 6.5
(12× 12, 100) 5.7 5.7
(6,6), (6, 8)(3× 3, 100) — —(6× 6, 100) 8.1 8.1
(12× 12, 100) 8.1 8.1
(7,7), (6, 8)(3× 3, 100) — —(6× 6, 100) 8.2 8.2
(12× 12, 100) 8.2 8.5
collocation form is applied on the spatial operators, and the resulting system of1
ODEs are solved by a new spectral multi-implicit spectral deferred correction2
(MISDC) method.3
Two flux reconstructions have been developed for discretizing the hy-4
perbolic terms: (i) extended element-wise flux reconstruction and (ii) non-5
extended element-wise flux reconstruction. For the hyperbolic terms, a low-6
order I-stable building block time integration scheme is introduced which leads7
to a stable and spectrally accurate space-time method when applying with the8
MISDC iteration process. In the new MISDC coupling method, the provisional9
solution is computed by the operator splitting method, and then this solution10
is corrected by a series of deferred correction steps; two different deferred cor-11
Multi-implicit space-time spectral element method 33
Table 6 Order of convergence in time of the present multi-implicit space-time spectralelement method for two different coupling strategies: coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind. Different combinationsof temporal polynomial order p(t) and the number of iterations K are investigated. Thetolerance is set to be 10−13, and the final time is T = 1.0.
Order of convergence in time
(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) Coupling 1 Coupling 2
(8, 8), (5, 4)(5× 5, 40) — —(5× 5, 80) 4.0 4.0(5× 5, 160) 4.0 4.0
(8, 8), (6, 4)(5× 5, 40) — —(5× 5, 80) 4.0 4.0(5× 5, 160) 4.0 4.0
(8, 8), (5, 5)(5× 5, 40) — —(5× 5, 80) 4.9 4.9(5× 5, 160) 5.0 5.0
(8, 8), (4, 5)(5× 5, 40) — —(5× 5, 80) 5.0 5.0(5× 5, 160) 5.0 5.0
(9, 9), (5, 6)(5× 5, 40) — —(5× 5, 80) 6.1 6.1(5× 5, 160) 6.0 6.0
(9, 9), (6, 6)(5× 5, 40) — —(5× 5, 80) 6.0 6.0(5× 5, 160) 5.9 5.9
rection strategies have been studied. Compared to low order operator splitting1
methods that each process is discretized in isolation, the splitting error in the2
MISDC method is eliminated by exploiting an iterative coupling strategy in the3
deferred correction procedure. Since the low order I-stable building block time4
integration scheme is a two-stage scheme, two correction strategies, “intrusive5
correction” and “non-intrusive correction,” are introduced and compared with6
each other. In the “non-intrusive” correction strategy, the correction terms7
are only applied on the last stage of the low-order I-stable scheme. Thus, the8
“non-intrusive” correction strategy requires less memory and is faster than the9
“intrusive strategy.”10
Numerical tests in two spatial dimensions are presented to demonstrated11
the performance of the present multi-implicit space-time spectral element12
method. The spectral convergence in both space and time is demonstrated13
for advection-diffusion-reaction problems. Both treatments for the hyperbolic14
terms exhibit spectral accuracy in both space and time. The accuracy of the15
extended element-wise flux reconstruction was found to be superior to the non-16
extended element-wise flux reconstruction with the same number of degrees of17
freedom. Both MISDC coupling strategies that we investigated were found to18
be spectrally accurate in both space and time.19
With the new coupling strategies, our new spectrally accurate space-time20
MISDC method can extend previous spectrally accurate in space and low or-21
der in time operator splitting methods to be spectrally accurate space-time22
numerical schemes. We envision that our spectral space-time method will be23
34 Chaoxu Pei et al.
of benefit to the engineering community because very high spatial and tem-1
poral resolution is needed for giving a correct description of the flow physics2
for the simulation of turbulent flows. Future work will concern the extension3
of the present space-time spectral element schemes to a hierarchical block4
structured space-time spectral element method to solve multiphase incom-5
pressible/compressible Navier-Stokes equations.6
Acknowledgements This work and the authors were supported in part by the National7
Science Foundation under contract DMS 1418983.8
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