Chaos, Solitons and Fractals 102 (2017) 327–332

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Fractional spectral vanishing viscosity method: Application to the

quasi-geostrophic equation

�

Fangying Song, George Em Karniadakis ∗

Division of Applied Mathematics, Brown University, 182 George St, Providence RI, 02912, USA

a r t i c l e i n f o

Article history:

Received 11 January 2017

Revised 20 March 2017

Accepted 24 March 2017

Available online 10 April 2017

Keywords:

Fractional conservations laws

Spectral element method

Singular solutions

a b s t r a c t

We introduce the concept of fractional spectral vanishing viscosity (fSVV) to solve conservations laws

that govern the evolution of steep fronts. We apply this method to the two-dimensional surface quasi-

geostrophic (SQG) equation. The classical solutions of the inviscid SQG equation can develop finite-time

singularities. By applying the fSVV method, we are able to simulate these solutions with high accuracy

and long-time integration with relatively low resolution. Numerical diffusion in fSVV can be tuned by

the fractional order as needed. Hence, fSVV can also be applied to integer-order conservation laws that

exhibit steep solutions and evolving fronts.

© 2017 Elsevier Ltd. All rights reserved.

1

J

s

a

b

u

b

t

∂

w

o

t

b

T

w

a

v

t

F

w

p

q

p

s

e

t

o

a

[

m

t

n

t

a

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t

s

t

i

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0

. Introduction

The general 3D quasi-geostrophic equations, first derived by

.G. Charney in the 1940s [1,2] , have been very successful in de-

cribing major features of large-scale motions in the atmosphere

nd the oceans in the mid-latitudes [3,4] . These 3D equations can

e reduced to the surface quasi-geostrophic (SQG) equation with

niform potential, modeling the potential temperature on the 2D

oundaries [5,6] . This paper presents a new numerical method for

he SQG equation

t θ + u · ∇θ + κ(−�) αθ = 0 , x = (x, y ) ∈ �, (1.1)

here κ ≥ 0 and α > 0 are parameters, � ∈ R

2 is a bounded peri-

dic domain, θ ( x , t ) is a scalar representing the potential tempera-

ure, and u = (u 1 , u 2 ) is the velocity field determined from θ ( x , t )

y the stream function ψ( x , t ) via the auxiliary relations

(u 1 , u 2 ) = (−∂ y ψ , ∂ x ψ ) , (−�) 1 2 ψ = −θ . (1.2)

he fractional Laplacian (−�) α in this paper is defined as follows

(−�) αθ ( x , t) =

∞ ∑

i =1

λαi c i (t) φi ( x ) , (1.3)

here (λi , φi ) ∞

i =1 are the eigenpairs of the standard Laplacian −�

nd θ has the expansion θ ( x , t) =

∑ ∞

i =1 c i (t) φi ( x ) . Alternatively,

� This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conser-

ation Laws and Beyond: Theory, Numerics and Applications (W911NF-15-1-0562)”. ∗ Corresponding author:

E-mail addresses: george_karniadakis@brown.edu , gk@dam.brown.edu

(G.E. Karniadakis).

(

b

s

h

a

d

ttp://dx.doi.org/10.1016/j.chaos.2017.03.052

960-0779/© 2017 Elsevier Ltd. All rights reserved.

he fractional operator (−�) α can be defined [7] through the

ourier transform

(−�) αθ (ω) = ω

2 α θ (ω) , (1.4)

here θ is the Fourier transform of θ [8] . When the fractional

ower α =

1 2 , the equation (1.1) derived from the more general

uasi-geostrophic models [9] describes the evolution of the tem-

erature on the 2D boundary of a rapidly rotating half-space with

mall Rossby and Ekman numbers. Dimensionally, the 2D SQG

quation with α =

1 2 is the analogue of the 3D Navier–Stokes equa-

ions. A general fractional order α is considered here in order to

bserve the minimal power of Laplacian necessary in the analysis

nd thus make a comparison with the 3D Navier–Stokes equations

10,11] .

The inviscid SQG Eq. (1.1) (i.e., κ = 0 ) is useful in modeling at-

ospheric phenomena such as the frontogenesis i.e., the forma-

ion of strong fronts between masses of hot and cold air [5,9] . The

umerical experiments show that the solution of the SQG equa-

ion with κ = 0 or κ � 1 emanating from very smooth initial data

ppears to exhibit the most singular behavior [5,12,13] . Since the

olutions will develop finite-time singularities, very high resolu-

ion is required for simulations in long time intervals [14] , making

uch computation very expensive. In this paper, we first introduce

he fractional spectral vanishing viscosity (fSVV) method for solv-

ng the SQG equation in cases of inviscid ( κ = 0 ) and inviscid-limit

κ � 1). The classical spectral vanishing viscosity (SVV) appears to

e effective in controlling solution monotonicity while preserving

pectral accuracy. It was initially developed for the resolution of

yperbolic equations using standard Fourier spectral methods [15] ,

nd later extended to large eddy simulation (LES) [16] . The stan-

ard SVV method has also been used for high Reynolds number

328 F. Song, G.E. Karniadakis / Chaos, Solitons and Fractals 102 (2017) 327–332

Table 1

Kinetic Energy and Helicity for the initial data θ 0 with β = 0 . 45 .

t L 2 -error K ( θ ) H ( θ ) L 2 -error (fSVV) K ( θ ) (fSVV) H ( θ ) (fSVV)

1 9.1056e −6 14.8044 26.7181 6.2618e −6 14.8044 26.7181

5 3.8480e −5 14.8044 26.7181 3.8481e −5 14.8044 26.7181

10 7.6486e −5 14.8044 26.7181 7.6485e −5 14.8044 26.7181

15 1.1459e −4 14.8044 26.7181 1.1459e −4 14.8044 26.7181

20 1.5273e −4 14.8044 26.7181 1.5273e −4 14.8044 26.7181

100 7.6328e −4 14.8044 26.7181 7.6327e −4 14.8044 26.7181

Table 2

Numerical results for the inviscid SQG equations, β = 0 . 45 .

t K ( θ ) H ( θ ) K ( θ ) (fSVV) H ( θ ) (fSVV)

1 14.804407 26.718074 14.804407 26.718074

3 14.804407 26.718074 14.804407 26.718074

5 14.804407 26.718074 14.804199 26.718054

8 14.923095 26.719398 14.775744 26.715950

10 NaN NaN 14.751368 26.714122

15 NaN NaN 14.532270 26.695471

20 NaN NaN 14.388240 26.6 814 87

Table 3

Parameters used for SQG with fSVV.

Case I Case II Case III Case IV Case V Case VI

m N N / 2 2 N / 3 N / 2 2 N / 3 N / 2 2 N / 3

εN 1 N

1 N

1 N

1 N

1 N

1 N

β 0.8 0.8 1.0 1.0 1.2 1.2

w

t

a

w

λ

H

[

E

c

n

�

i

w

l

f

θ

T

a(

u

incompressible flows [17,18] and for the fractional Burgers equation

[19] . Following the fractional Laplacian (−�) α defined in Eq. (1.3) ,

we define a new fSVV operator S βN based on a similar eigenfunc-

tion expansion; the exact formula of S βN will be given in the next

section. This operator plays an important role in stabilizing the

high frequency modes of the numerical solution.

The remainder of this paper is organized as follows. In

Section 2 we show how to implement the fSVV method in the

framework of the spectral element approximation. We propose to

use an approximate form, which can be readily implemented in

existing solvers [20] . The advantage of such an approximate form

is that the computational cost per time-step is roughly the same

with and without fSVV stabilization. In Section 3 we present the

numerical results. A brief study of the influence of the fSVV tuning

parameters on the convergence and accuracy is provided. Then, we

consider the inviscid and viscous SQG equation with smooth initial

conditions, and investigate systematically the effectiveness of the

fSVV method. Finally, we provide a short summary in Section 4 .

2. Numerical method

In previous work, we have developed a numerical method

for computing fractional Laplacians on complex-geometry domains

[20] , by considering the following Eigen Value Problem (EVP) for

the Laplacian:

− �u − λu = 0 , x ∈ �, (2.1)

proper boundary conditions. (2.2)

For the problems we consider here we will employ peri-

odic boundary conditions but in principle any Dirichlet and Neu-

mann boundary conditions can be applied. The spectral element

method (SEM) [21,22] is used for solving Eqs. (2.1) and (2.2) . Then,

Eqs. (2.1) and (2.2) can be written in the discretized form

A N U − λM N U = 0 , (2.3)

where N represents the number of the degrees-of-freedom (DoF)

of the linear system (2.3) for the given number of elements El

and polynomial degree N in each element. A N is the correspond-

ing matrix of the Laplacian operator under certain boundary con-

ditions, M N is the mass matrix, and U is the numerical solution

of u . The continuous EVP is approximated by the numerical so-

lution of the eigenpairs (λi , φi ) N i =1

of the matrix K = M

−1 N A N , and

λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λN . Using the numerical eigenpairs ( λi , φi ) of the Laplace opera-

tor −� from the SEM solution, we can approximate the fractional

Laplace operator as

(−�) αu ≈N ∑

i =1

c i λαi φi , (2.4)

where c i = (φi , u ) N , and (·, ·) N represents discrete inner product

base on Gaussian quadrature in every element. The numerical re-

sults show that this method is converging exponentially for frac-

tional diffusion equation with smooth solutions. However, the clas-

sical solutions of the SQG equation can develop finite-time singu-

larities with smooth initial conditions [14] . Due to this problem, we

introduce a tunable fractional spectral vanishing viscosity (fSVV)

method for solving the SQG equation. The variational statement of

the problem reads as θN , ψ N ∈ V N (�) , ∀ v ∈ V N (�) so that (∂θN ∂t

, v )

N + ( u N · ∇θN , v ) N + κ

((−�N )

αθN , v )N

+ εN (S βN θN , v

)N = 0 , (2.5)

(u N , 1 , u N , 2

)=

(− ∂ y ψ N , ∂ x ψ N

), (−�N )

1 2 ψ N = −θN , (2.6)

B

here εN = O ( 1 N ) , V N (�) = span { φi , i = 1 , . . . , N } and β > 0 is a

unable fractional order. The fractional operators (−�N ) α and S βN

re defined as follows

(−�N ) αu =

N ∑

i =1

λαi u i φi , S

βN u =

N ∑

i =1

˜ λβi

u i φi , (2.7)

here u i = (u, φi ) N and

˜ i =

{0 , i ≤ m N ,

exp (−( d−i m N −i

) 2 ) λi , i > m N . (2.8)

ere m N can have different forms m N = { √

N , N 2 , or 2 N 3 etc. }

16,23] . Of course, the usual spectral approximations of

qs. (1.1) and (1.2) are recovered when εN = 0 or m N = N .

Next, the ordinary differential Eqs. (2.5) and (2.6) are dis-

retized by the second-order Crank–Nicolson scheme. Let L be the

umber of the time steps to integrate up to final time T , then

t = T /L . We denote by superscripts the time levels and set the

nitial condition θ0 N = θN ( x , 0) and θ−1

N = θ0 N . Here, we simulate

ith a first-order scheme in the first time step. We look for so-

ution of (θn +1 N , u

n + 1 2

N , ψ

n + 1 2

N ) for n = 0 , . . . , L − 1 . We introduce the

ollowing notation for convenience:

n + 1 2

N =

1

2

(θn +1 N + θn

N ) , θ∗,n + 1 2

N =

1

2

(3 θn N − θn −1

N ) .

hen, the fully discrete scheme of the SQG equation can be written

s follows

θn +1 N − θn

N �t

, v )

N + ( u

n + 1 2

N · ∇θ∗,n + 1 2

N , v ) N

+ κ((−�N )

αθn + 1 2

N , v )N + εN

(S βN θ

n + 1 2

N , v )N = 0 , (2.9)

n + 1 2

N =

(− ∂ y ψ

n + 1 2

N , ∂ x ψ

n + 1 2

N ), (−�N )

1 2 ψ

n + 1 2

N = −θ∗,n + 1 2

N . (2.10)

y the orthogonality of the eigenfunctions we obtain

F. Song, G.E. Karniadakis / Chaos, Solitons and Fractals 102 (2017) 327–332 329

(a) Case I. (b) Case II. (c) Case III. (d) Case IV. (e) Case V. (f) Case VI.

Fig. 1. Evolution of ellipse with eccentricity of 4 in SQG. θ contours are shown at times t = 8 , 16 , 26 , 35 from top to bottom.

Fig. 2. The energies K ( θ ( t )) and H ( θ ( t )) versus t .

Table 4

The decay rates at t = 40 .

Case I Case II Case III Case IV Case V Case VI

R K (%) 2.681 2.269 2.567 1.497 2.296 0.904

R H (%) 0.066 0.041 0.062 0.022 0.055 0.008

β 0.8 0.8 1.0 1.0 1.2 1.2

c

h

w

3

t

S

p

3

m

e

∂

w

a

θ

T

( 1 2

n +1 i

− c n i + �t( u

n + 1 2

N · ∇θ∗,n + 1 2

N , φi ) N

+

�t

2

(κλα

i + εN λβi

)(c n +1

i + c n i ) = 0 , (2.11)

λ1 2

i p

n + 1 2

i = −c

∗,n + 1 2

i , (2.12)

ere ψ

n + 1 2

N ( x ) has the expansion ψ

n + 1 2

N ( x ) =

∑ N i =1 p

n + 1 2

i φ( x ) , then

e compute u

n + 1 2 = (−∂ y ψ

n + 1 2 , ∂ x ψ

n + 1 2 ) .

N N N

. Numerical results

We test the accuracy and efficiency for the inviscid SQG equa-

ion (i.e., κ = 0 ) in the first subsection. We consider the dissipative

QG equation (i.e., κ > 0) in the second subsection. All the exam-

les are considered in a periodic domain.

.1. The inviscid SQG equation

We now test the accuracy and efficiency of the numerical

ethod developed in Section 2 . We focus on the inviscid SQG

quation

t θ + u · ∇θ = 0 , θ ( x , 0) = θ0 ( x ) , (3.1)

ith u = (U 1 , U 2 ) being a constant velocity. This transport equation

dmits the explicit solution

(x, y, t) = θ0 (x − U 1 t, y − U 2 t) .

he numerical method is implemented and the solution of

3.1) is computed with U = 1 , U = 1 and the initial condition

330 F. Song, G.E. Karniadakis / Chaos, Solitons and Fractals 102 (2017) 327–332

Fig. 3. ‖∇θ ( t ) ‖ ∞ versus t .

c

t

t

l

i

m

e

θ

W

N

i

i

s

t

T

t

c

c

c

e

c

R

R

W

C

a

f

i

g

t

3

[

1

θ0 ( x ) = sin (x ) sin (y ) + cos (y ) . We set N = 7 , El = 400 , �t =10 −3 , � = (0 , 2 π) 2 , β = 0 . 45 , m N =

√

N and εN = 1 /N. The DoF

N = 141 × 141 in this test. The Eq. (3.1) satisfies the two basic con-

servation laws, namely the conservation of the Kinetic Energy and

of the Helicity

K(θ ) =

1

2

∫ �

θ2 ( x , t) d x =

1

2

∫ �

θ2 0 ( x ) d x ,

H(θ ) = −∫ �

ψ( x , t) θ ( x , t) d x = −∫ �

ψ 0 θ0 d x .

(3.2)

Table 1 lists the errors between the exact solution and the com-

puted solution at several times. The results show both the stan-

dard spectral method (i.e., εN = 0 ) and the fSVV method are stable

and accurate in this case. This implies that the fSVV term does not

affect the numerical solution converging to the analytic solution.

Remark 3.1. We use the Crank–Nicolson scheme for time dis-

cretization. It is second-order scheme C �t 2 , where C depends on T,

and hence we see that the errors increase by two orders of mag-

nitude at later times.

Next, we compute solutions of the inviscid SQG equation with

initial data given as θ0 ( x ) = sin (x ) sin (y ) + cos (y ) and with the

same parameters in the previous case. Table 2 records the numeri-

Fig. 4. Contours of

al values of these quantities at various times. The results indicate

hat our algorithm is convergent and stable for the nonlinear equa-

ions. Moreover, the spectral element method without fSVV stabi-

ization will blow up for large time simulation. This phenomenon

s also presented in [5,12] , for the standard spectral collocation

ethod with insufficient resolution.

For the last numerical test of SQG flow in this subsection, we

mploy the following initial condition describing a vortex:

0 ( x ) = exp

(− (x − π) 2 − 16(y − π) 2

). (3.3)

e solve the system (1.1), (1.2) with κ = 0 , � = [0 , 2 π ] 2 . We set

= 7 , El = 2500 , �t = 10 −3 , h = π/ 25 . The DoF N = 351 × 351

n this test. We show the parameters for SQG equation with fSVV

n Table 3 .

Fig. 1 shows the numerical results for inviscid equation for the

mooth elliptical initial condition (3.3) for several cases. We see

hat filaments are shed by the spinning vortex in a circular shape.

hese visualizations are consistent with results presented in [6] for

ime up to t = 16 for all cases. By t = 26 , 40 , instabilities are

learly seen along the filaments, with other small-scale structure

loser to the vortex. Fig. 2 plots the energies versus time for the

ases corresponding to the parameters in Table 3 . We find that the

nergies decay slightly by adding the fSVV term. We define the de-

ay rate as follows

K (t) = (1 − K (θN (t)) /K (θ0 )) ∗ 100% ,

H (t) = (1 − H (θN (t)) /H (θ0 )) ∗ 100% .

e show the energy decay rates in Table 4 corresponding to the

ase I − V I at t = 40 . The energy can be conserved better by using

higher fractional order β and fewer high frequency modes in the

SVV term m N . Finally, we plot the maximum gradient ‖∇θN (t) ‖ ∞

n Fig. 3 . As indicated in [5] , the maximum gradient ‖∇θN (t) ‖ ∞

rows as the time evolves. Moreover, it grows rather rapidly after

= 7 . Due to the instabilities, ‖∇θN (t) ‖ ∞

oscillates for t ∈ [16, 40].

.2. The dissipative SQG equation

We solve the system (1.1), (1.2) with α = 0 . 4 , κ = 0 . 001 , � =0 , 2 π ] 2 . We set N = 7 , El = 2500 , �t = 10 −3 , h = π/ 25 , εN = /N and m N =

√

N with several β = 0 . 4 , 0 . 425 , 0 . 45 , 0 . 50 . The

θ , here t = 6 .

F. Song, G.E. Karniadakis / Chaos, Solitons and Fractals 102 (2017) 327–332 331

Fig. 5. Contours of θ , here t = 8 .

Fig. 6. Contours of θ , here t = 16 .

Fig. 7. Contours of θ , here t = 20 .

332 F. Song, G.E. Karniadakis / Chaos, Solitons and Fractals 102 (2017) 327–332

t

[

[

[

DoF N = 351 × 351 in this test. We compute the solution corre-

sponding to the initial condition given as

θ ( x , 0) = sin (x ) sin (y ) + cos (y ) . (3.4)

Figs. 4–7 plot the evolution of the contours of θ from t = 6 to

= 20 for several β . We obtain similar contours as the results of

Fig. 29 in reference [14] but at much less resolution (the DoF in

reference [14] is 2048 × 2048). Especially, the case for β = 0 . 45 is

very close to the numerical results in reference [14] . The spectral

element method without fSVV stabilization (i.e., εN = 0 ) blows up

again for solving inviscid-limit SQG equation after time up to 9.

4. Summary

We have extended the spectral vanishing viscosity to fractional

conservation laws using a simple definition of the fractional Lapla-

cian for an easy implementation in high dimensions. In particu-

lar, by introducing the fractional order β we can better control the

diffusion operator rather than relying on the artificial viscosity εN and the mode cut off parameter m N . Here we applied the method

for the SQG equation but ongoing tests with other conservation

laws, e.g. nonlocal Burgers equation [24] , also show the effective-

ness of fSVV. We note that fSVV can also be applied to integer-

order conservation laws that exhibit steep solutions. The fSVV

method can be extended to time and space fractional reaction-

diffusion equations with the solutions develop finite-time singu-

larities directly by using an efficient time scheme [25–27] .

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