high order shock detectors in hybrid weno-compact finite ... · high order shock detectors in...

High Order Shock Detectors in Hybrid

WENO-Compact Finite Difference Scheme for

Hyperbolic Conservation Laws

Advances in PDEs: Theory, Computation & Application toCFD

ICERM, Brown University

Providence, RI, USA

Wai Sun, DONSchool of Mathematical Sciences

Ocean University of China, Qingdao, Shandong, China

August 20 - August 25, 2018

Saul Abarbanel

Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme – 2 / 56

http://www.ouc.edu.cn/

Saul Abarbanel


Brown (1988)

ICASE (1989- ) in Summer

Brown (1988-) whenever he came calling to visit David

Conferences, meetings and panels

A Mentor

A Collaborator

A Friend


Saul Abarbanel


Secondary frequencies in the wake of a circular cylinder with vortexsheddingJournal of fluid mechanics, 1991Saul S Abarbanel, Wai Sun Don, David Gottlieb, David H Rudy, James CTownsend

The Theoretical Accuracy of Runge-Kutta Time Discretizations for theInitial Boundary Value Problem: A Careful Study of the BoundaryConditionSIAM JSC, 1993Carpenter, Mark H ; Gottlieb, David ; Abarbanel, Saul ; Don, Wai-Sun


Saul Abarbanel


Copyrighted by Disney.


Talk Overview


Hybrid Scheme for Nonlinear Hyperbolic PDE with Shocks

High Order Nonlinear WENO Finite Difference Scheme

High Order Linear Compact Finite Difference Scheme

High Order shock detection algorithms, based on

Multi-resolution analysis

Conjugate Fourier analysis

Multi-Quadric Radial Basis Function analysis

Numerical results


Nonlinear Hyperbolic PDEFinite Time Singularity – Shock


∂Q

∂t+∇ · F(Q) = S. Burgers/Euler equations

LINEAR scheme: The well-known Gibbs oscillations appears around thelocation of the shock (Fourier spectral method, Compact scheme ).

NON-LINEAR Scheme: Numerical or physical based adaptive dissipationvia upwinding or localized regularization(TVD, ENO, WENO, etc.)

x

Q

0 1 2 3 4 5 6-1

-0.5

0

0.5

1


Sufficient Condition for Formal Order of

WENO FD scheme


The sufficient condition for the optimal order of the WENO finite differencescheme for the hyperbolic PDEs is

∆ω±

k = ω±

k − dk = O(∆xr), (1)

It measures the deviation of the nonlinear ωk from the ideal weights dk.

LARGE ⇒ Lower order; Upwind; Larger Dissipation; Shock Capturing

SMALL ⇒ High order; Central; Lesser Dissipation; Higher Resolution

A WELL-BALANCED definition of ωk is essential in

capturing shocks essentially non-oscillatory; while

resolving smooth solution.


The Classical WENO-JS Scheme


The lower order local smoothness indicators

βk = ∆x

∫ xi+1

2

xi−

12

(d

dxqk(x)

)2

dx+∆x2∫ x

i+12

xi−

12

(d2

dx2qk(x)

)2

dx, (2)

measures the sum of a normalized L2 norm of the derivatives of the seconddegree polynomials qk(x) in substencil Sk in cell xi = [xi− 1

2

, xi+ 1

2

].

The nonlinear weights of the classical WENO-JS scheme (Jiang and Shu) are

αk =dk

(βk + ε)p, ωk =

αk∑2l=0 αl

,

with a user defined power parameter p = 2 and the sensitivity parameterε = 10−6, 10−40, a fixed value, to avoid the division of zero when βk = 0.


Smoothness Indicators


In terms of cell averages values of f(x), fi, one has

β0 =13

12(fi−2 − 2fi−1 + fi)

2 +1

4(fi−2 − 4fi−1 + 3fi)

2,

β1 =13

12(fi−1 − 2fi + fi+1)

2 +1

4(fi−1 − fi+1)

2,

β2 =13

12(fi − 2fi+1 + fi+2)

2 +1

4(3fi − 4fi+1 + fi+2)

2.

Their Taylor series expansions at xi are

β0 = f′2

i ∆x2 +

(

13

12f

′′2

i −

2

3f′

if′′′

i

)

∆x4−

(

13

6f′′

i f′′′

i −

1

2f′

if′′′′

i

)

∆x5 +O(∆x6),

β1 = f′2

i ∆x2 +

(

13

12f

′′2

i +1

3f′

if′′′

i

)

∆x4 +O(∆x6),

β2 = f′2

i ∆x2 +

(

13

12f

′′2

i −

2

3f′

if′′′

i

)

∆x4 +

(

13

6f′′

i f′′′

i −

1

2f′

if′′′′

i

)

∆x5 +O(∆x6).


The Improved WENO-Z Scheme


By introducing the optimal global smoothness indicator,

τ5 = |β0 − β2|(+

(13

3f

′′

i f′′′

i − f′

if′′′′

i

)∆x5 +O(∆x6)

).

One can define the nonlinear weights

αzk = dk

(1 +

(τ2r−1

βk + ε

)p), ωz

k =αzk∑r−1

l=0 αzl

.

τ5 measures the higher derivatives (smoothness) of f(x) on S5, if existed.

βk is of order O(∆x2).

τ5 is of order O(∆x5).

τ2r−1

βkis of order O(∆xr), satisfying the sufficient condition

δωk = O(∆xr) for optimal order provided that f ′ 6= 0.


Loss of Formal Order of WENO scheme

at Critical Points


Critical points of order ncp : f ′(0) = . . . = f (ncp)(0) = 0 and f (ncp+1)(0) 6= 0).

The nonlinear weights of the WENO-JS and WENO-Z schemes ωk do not satisfythe sufficient conditions in the present of critical points of order (ncp ≥ 1) forWENO-JS and (ncp ≥ 2) for WENO-Z, for a fixed small ε.

∆x WENO-JS7 WENO-Z7 (p = 2) WENO-Z7 (p = 3)1.00000E-01 1.2E-03 4.1E-04 8.1E-04 0.05.00000E-02 3.0E-05 5.3 2.8E-06 7.2 3.4E-06 7.92.50000E-02 8.6E-07 5.1 1.8E-08 7.3 7.7E-09 8.81.25000E-02 2.6E-08 5.1 1.3E-10 7.1 2.3E-11 8.46.25000E-03 8.0E-10 5.0 1.0E-12 7.0 8.3E-14 8.13.12500E-03 4.0E-11 4.3 1.2E-14 6.4 3.4E-16 7.91.56250E-03 2.2E-12 4.2 1.9E-16 6.0 2.7E-18 7.07.81250E-04 1.3E-13 4.1 2.9E-18 6.0 2.1E-20 7.0

Table I: Rate of convergence at a second order critical point (ncp = 2) forthe seventh order (r = 4) WENO scheme with ε = 10−40 .


Numerical Dissipation of WENO Scheme


Entropy with N = 1500 at time t = 5.

x

Ent

ropy

3 4 5 6 7 8 90.42

0.43

0.44

0.45

0.46

0.47

0.48

ReferenceFC-WENO-Z5WENO-Z5

x

Ent

ropy

4 4.1 4.2 4.3 4.4 4.5 4.60.43

0.44

0.45

0.46

0.47

ReferenceFC-WENO-Z5WENO-Z5


Disadvantages of using WENO Scheme


The WENO-JS/Z schemes, in order to guarantee essentially non-oscillatory atHIGH GRADIENTS, in spite of the improvements made, is

4–5 times more expensive compared to other nonlinear schemes,Flux-Splitting, Roe Eigensystem, Characteristic Forward and

Backward Projections, Smoothness Indicators, Nonlinear Weights,are needed to be computed at each grid points (Cooks et al.)regardless of the solution is smooth or not.

Dissipative and Dispersive,

Degradation of accuracy for functions with critical points,

Given so many different upgraded WENO schemes (MP-WENO,WENO-M, C-WENO, ES-WENO, WENO-η, WENO-NS, WENO-CU,TENO, WENO-Z+, ....), the ecosystem becoming increasing difficult tofigure out the right version of WENO scheme to use and parameters totune for a given scheme and problem.


Space-Time Adaptive Hybrid scheme


A natural remedy to alleviate some of these difficulties is to AVOID usingWENO NONLINEAR scheme at the known SMOOTH regions of thesolution at a given time.

Hybridization of the high order WENO nonlinear scheme for discontinuoussubstencils (subdomains) and varieties of high order linear schemes for smoothsubstencils (subdomains) have been devised.

Example: Hybrid Compact-WENO scheme (Pirozzoli, Ren, etc.)


Hybrid Scheme


High Order linear finite difference schemes can be

Central scheme in single-/multi-domain framework :(local, general, simple, non-dissipative, dispersive, high formal order andefficient.)

Band-width optimized scheme in single-/multi-domain framework :(local, simple, non-dissipative, high resolution, lower formal order,efficient, but requires a good choice of user defined parameters.)

Compact scheme in single-/multi-domain framework :(global, relatively simple, non-dissipative, higher resolution, but requiresboundary closure.)

——————————————————————————


Hybrid Scheme


High Order linear spectral schemes can be

Chebyshev collocation method in multi-domain framework :(global, relatively simple, non-dissipative, non-dispersive, spectralaccuracy, FFT, non-uniform grid, require interpolation betweensubdomains and small time step ∆t.)

Fourier-Continuation method in single-/multi-domain framework :(global, complicated, non-dissipative, non-dispersive, spectral accuracy,uniform grid, FFT, large time step ∆t, require an overlap betweensubdomains, solution of a severe to mild ill-condition system and a carefulchoice of user defined parameters.)


High Order Compact scheme


A 6th-order (cr = 6) compact finite difference scheme on a uniformly spacedgrids, with c = 1

36∆x,

Ag′ = cBg + b,

A =

1 1/31/3 1 1/3

. . .. . .

. . .

1/3 1 1/31/3 1

,B =

0 28 1−28 0 28 1−1 −28 0 28 1

. . .. . .

. . .. . .

. . .

−1 −28 0 28−1 −28 0

.

b = c (−g−1 − 28g0,−g0, 0, · · ·, 0, gN , 28gN + gN+1)T − 1

3(g′0, 0, 0, · · ·, 0, 0, g′N)

T,

where g−1 = g(x0 −∆x) and gN+1 = g(xN +∆x) are the ghost points.g′0 and g′

Nare computed by the WENO-Z scheme.


Space-Time Adaptive Hybrid WENO

Finite Difference scheme


1. Perform the shock detection analysis on one or more suitable variable(s)once at the beginning of a time stepping scheme.Example, multi-resolution, Conjugate Fourier, IAMQ-RBF-Fast.

2. Set a WENO Flag using a shock detection algorithm .

3. A buffer zone with m points is created around each grid point xi that allthe grid points inside the buffer zone are flagged as non-smooth stencils.

This condition prevents the computation of the derivative of the fluxes bythe compact scheme using non-smooth function values.

4. Compute the derivative of the fluxes at each cell center by

(Non-smooth stencil) : Use the WENO scheme.

(Smooth stencil) : Use the compact scheme.


High Order Smoothness analysis


A successful implementation of a high order Hybrid scheme strongly dependson the ability to obtain HIGH ORDER accurate information on thesmoothness of and sharp detection of high gradients/shocks/edges in thesolution of the PDEs.

We seek a shock detection algorithm, if existed, that is

1. ACCURATE (pinpointing shock locations),

2. FAST (CPU time), and

3. ROBUST (ease of parameters tuning).

To detect the smooth and rough parts of the solution on a equi-distant gridwith high resolution/accuracy, one has the

1. Lagrange polynomial based Multi-Resolution (MR) analysis,

2. Trigonometric polynomial based conjugate Fourier (cF) analysis,

3. Multi-Quadric Radial Basis function (IAMQ-RBF) based edgedetection analysis.


Multi-resolution Analysis (Harten)


Given an initial number of the grid points N0 and grid spacing ∆x0, we shallconsider a set of nested dyadic grids up to level L < log2N0,

Gk = xki , i = 0, . . . , Nk, 0 ≤ k ≤ L, (3)

where xki = i∆xk with ∆xk = 2k∆x0, Nk = 2−kN0 and the cell averages offunction u at xki :

uki =

1

∆xk

∫ xk

i

xk

i−1

u(x)dx. (4)

Define the 2s = q− 1 degree POLYNOMIAL

uk2i−1 (5)

that approximates uk2i−1 interpolating uk

i+l, |l| ≤ s at xki+l.


Multi-resolution analysis (Cont.)


The approximation error (or multi-resolution coefficients),

di = u02i−1 − u0

2i−1, (6)

at xi, has the property that if u(x) is a Cp−1 function, then

di ≈

[u(p)i ] ∆xp p ≤ q

u(q)i ∆xq p > q

, (7)

where [·] and (·) are the jump and the derivatives of the function, respectively.

Controlled by the

1. smoothness of the underlining function p,

2. degree of the interpolation polynomial q = 2s+ 1.




x

MR

Coe

ffici

entd

f(x)

-0.5 0 0.510-12

10-10

10-8

10-6

10-4

10-2

100

-2

0

2

4

6

8

10

12

MR 3rd OrderMR 8th Orderf(x)

x

MR

Coe

ffici

entd

(x)

f(x)

-4 -2 0 2 410-10

10-8

10-6

10-4

10-2

100

0

1

2

3

4

5f(x)3rd Order5th Order7th Order

Figure 1: (Left) The third and eighth order MR coefficients di of a piecewiseanalytic function.(Right) The third, fifth and seventh order MR coefficients di of the densityρ(x) of the Mach 3 Shock-Entropy wave interaction problem.


2D Detonation Diffraction Problem :Hybrid-Compact-MR

Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme – 24 / 56——————————————————————————




The MR flags in the x− and y-directions of detonation diffraction around a90o corner with Nx ×Ny = 400× 400.

WENO Flag in x WENO Flag in y

——————————————————————————




Table II: Comparative CPU timing and speedup for the two-dimensional deto-nation diffraction problem.

2r − 1 cr Nx ×Ny WENO-Z Hybrid-Compact Speedup

5 6400× 400 1697 691 2.5

1000× 1000 27980 8430 3.3




Pros

Fast with Banded Matrix-Matrix Multiply operation.

Easy to implement.

Good accuracy in identifying location of high gradients/shocks.

Rapid decays in smooth regions as the order increases.

Cons

Require a parameter ǫMR (typical 10−3) to determine where the highgradients/shocks are.


Conjugate Fourier Analysis


Consider the power series on the unit circle with z = exp(ix),

F (z) =1

2a0 +

∞∑

k=1

(ak − ibk)zk, (8)

with the Fourier coefficients ak and bk.

Its REAL part is the infinite Fourier series of a periodical function f(x) :

f(x) =1

2a0 +

∞∑

k=1

(ak cos kx+ bk sin kx), (9)

Its IMAGINARY part is called the conjugate Fourier series (Zygmund

(1968), Lukacs (1920), Fejer (1913) etc.):

f(x) =∞∑

k=1

(ak sin kx− bk cos kx), (10)


Conjugate Fourier Analysis (Cont.)


Truncate the series into N degree trigonometric polynomials :

fN(x) =1

2a0 +

N∑

k=1

(ak cos kx+ bk sin kx), (11)

fN(x) =

N∑

k=1

(ak sin kx− bk cos kx). (12)

(fN)′(x) = fN(x) =

N∑

k=1

k(ak cos kx+ bk sin kx). (13)




f(x) fN(x) fN(x)

x

f(x)

0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

x

FC

0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x0.5 1 1.5 2 2.5 3

-30

-20

-10

0

10

20

30




The concentration property of fN(x) gives the location and jump of adiscontinuity based on the following Lemma1:

Lemma 1. Let f(x) be a 2π-periodic piecewise smooth function, except asingle jump discontinuity at x = ξ with an associated jump

[f ](ξ) := f(ξ+)− f(ξ−), (14)

then

− π

logNfN(x) → [f ](x)δξ(x) =

[f ](ξ), x = ξ,0, otherwise.

(15)

where δξ is the Dirac distribution located at ξ.

1Gelb and Tadmor, 97, 99




Pros

Fast if implemented with Fast Cosine/Sine Transform (FFT).

Sharp detection of the location of discontinuities.

Good approximation of the jump of a discontinuity can be obtain(Bonus).

Cons

More complex to implement.

Slower than MR analysis with a O(5N logN) floating pointsoperation.

The rate of convergence is only O(

1logN

)at best.

Require ǫcF to identify the local of the discontinuities.


Iterative Conjugate Fourier ShockDetection Algorithm


(Right) Iterative cF shock detector with N = 128.

(Left) Iterative cF shock detector with N = 256.


Two dimensional Riemann IVP problem

(Hybrid Compact-cF)


Rho Pressure

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

11.641.521.301.181.060.930.810.690.570.440.320.20

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.551.411.291.130.980.820.670.510.360.200.05


Two dimensional Riemann IVP problem

(Hybrid Compact-cF)


WENO Flag in x WENO Flag in y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


Multi-Quadric Radial Basis Function

(MQ-RBF) methods


φi(x) =√(x− xi)2 + εi2, xi ∈ X, x ∈ Ω, i = 1, 2, · · · , n.

The MQ-RBF approximation g(x) and its derivative for a f(x) ∈ R are

g(x) =

n∑

i=1

λiφi(x), g′(x) =

n∑

j=1

λj(x− xj)/φi(x). (16)

where λi are the RBF expansion coefficients.By defining the interpolation matrix M with Mij =

√(xi − xj)2 + εj2, one has

λ = M−1f , (17)

By defining the differentiation matrix D with Dij = (xi − xj)/Mij , one has

g′ = Dλ, (18)

Remark : M is a symmetric Toeplitz matrix on a uniformly spaced mesh.


Concentration Map and Edge Set


A normalized concentration map C, that reflects the smoothness of thefunction and indicates the possible locations of the local jumps, is defined by

C =

Ci‖Ci =

Ci

maxC, Ci = |Dλ|2, xi ∈ X, i = 1, 2, · · · , n

. (19)

The edge set S can then be defined

S = Si|Si = xi, ‖ Ci ≥ η, g′(xi) 6= 0, xi ∈ X, i = 1, 2, · · · , n, (20)

where η ∈ (0, 1) is the given tolerance parameter.


1D edge detection with IAMQ-RBF

method


A simple sign function is defined as

f(x) =

−1, −1 ≤ x < 0,1, 0 ≤ x ≤ 1.

(21)

Figure 2: Left : Edges detected by the IAMQ-RBF-Fast method withǫ = 0.1, η = 0.5 and n = 50.Right: The normalized concentration map C.


Iterative Adaptive Multi-Quadric Radial

Basis Function (IAMQ-RBF) methods


The iterative adaptive MQ-RBF (IAMQ-RBF) method is based on thegrowth/decay properties of the coefficients λ (Jung et al. 09). That is,λ grows (decay) exponentially near and at (away from) thediscontinuities.

The IAMQ-RBF method is iteratively

1. Set the constant shaper parameter εi = ǫ to find the edge set S fromthe concentration map C.

2. Iteratively/Adaptively set ǫi = 0 for those xi ∈ S to remove thedetection of the jumps at xi in the subsequent iterations.

3. STOP when the residual ||λk+1 − λk||2 ≤ δ, (δ = 10−8).


Fast solver for Toeplitz system in RBF

method


Problem : Step 2 requires to find the time consuming inverse of aperturbed NON-SYMMETRIC Toeplitz-ǫ matrix system of the form

Mx = (T+P)x = b

at each iteration.

M =

ε√h2 . . .

√(nh)2 + ε2

√h2 + ε2 0

. . ....

.... . .

. . .√h2 + ε2√

(nh)2 + ε2√((n− 1)h)2

√h2 + ε2 ε

Solution : Find a FAST algorithm!


Fast solver for Toeplitz system in RBF

method


In the iterative steps of the IAMQ-RBF method, the perturbed Toeplitz-ǫmatrices are generated and one need an efficient solver for the system

(T+P)x = b,

where T ∈ Rn×n, and P ∈ Rn×n is a zero matrix except its columnsn1, n2, · · · , nm and m ≪ n.

Write P = UVT where

U ∈ Rn×m with the columns being the non-zero columns of P, and

V ∈ Rn×m is the permutation matrix, that is,

V = (en1, en2

, · · · , enm) ∈ Rn×m,

with ei ∈ R1×n being the i-th unit column vector.


The solver for Toeplitz-ǫ matrix


By the Sherman-Morrison-Woodbury formula, one has

(T+UVT )−1 = T−1 −T−1U(Im +VTT−1U)−1VTT−1, (22)

where Im ∈ Rm×m is an identity matrix. Thus, we only need to solve twoToeplitz systems

TQ = U,Tv = b,

(23)

and a small (m×m) system

Bw = VTv, (24)

whereB = Im +VTQ ∈ Rm×m. (25)

to obtain the solutionx = v −Qw. (26)


Fast recursive O(n2) algorithm for T−1


NEED a Fast O(n2) direct solver for T−1

Instead of the (O(n3) classical Gaussian-Elimination method, we takeadvantage of the special structure of the Toeplitz matrix and to compute itsinverse recursively including

the O(n2) Levinson-Durbin algorithm (Trench 64),which also employed the

Yule-Walker algorithm.

Both of them are O(n2) recursion algorithms. (See reference for details)

We shall refer the algorithm for solving (T+P)x = b as IAMQ-RBF-Fastmethod.


1D edge detection with IAMQ-RBF-Fast

method


x

f(x)

-1 -0.5 0 0.5 1

0

5

10

15 f(x)L1PAMRIAMQ-RBFIAMQ-RBF-Fast

Figure 3: Final detected edges of the piecewise linear function by the IAMQ-RBF-Fast method with ǫ = 0.1, η = 0.5 and n = 50.



method


First Second Third Fourth

x

Fla

g

-1 -0.5 0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Fla

g

-1 -0.5 0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Fla

g

-1 -0.5 0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Fla

g

-1 -0.5 0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

C

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

C

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

C

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

C

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 4: The edges of the piecewise linear function detected by the IAMQ-RBF-Fast method at each iteration. Bottom: The normalized concentrationmap C at each iteration.



method


The final detected edges of Shepp-Logan image, sunflower image and three classicalphotos downloaded from USC Signal and Image Processing Institute Data base withthe size of 256× 256 are shown respectively.

Figure 5: Final edges of the Shepp-Logan image. Left: The original image.Right: Edges detected by the IAMQ-RBF-Fast method with ǫ = 0.1, η = 0.1.



method


Figure 6: Final edges of the sunflower image with the size of 256× 256. Left:The original image. Right: The detected edges by the IAMQ-RBF-Fast methodwith ǫ = 0.1, η = 0.4.



method


Figure 7: Final edges of the clock image with the size of 256× 256. Left: Theoriginal image. Right: The detected edges by the IAMQ-RBF-Fast methodwith ǫ = 0.1, η = 0.05.



method


The comparison with the original method on CPU timing shown in Table IIIdemonstrates that the present method is an effective (at least three timesfaster than the original IAMQ-RBF method) technique for edges detection.

Table III: The CPU timings (in second).

Test images RBF RBF-Fast Speedup

Sunflower image 2.44 0.44 5.5Clock image 2.44 0.69 3.6

Airplane image 2.32 0.28 8.1Shepp-Logan image 2.03 0.25 7.9Resolution test image 2.26 0.56 4.0


Edge Detection with IAMQ-RBF-Fast

method


Pros

Fast with the fast O(n2) direct solver.

Sharp detection of the location of discontinuities.

Fairly straightforward to implement.

Captures almost ALL discontinuities within a finite number ofiterations.

Cons

Only appropriate for uniformly spaced mesh.

Slightly slower than other edges (or shocks) detection algorithm.

Domain decomposition.

Require a parameter (η) to identify the locations of the discontinuities.

Tukey’s boxplot method.


Examples: 1D Shock Detection


Consider the extended Mach 3 shock density wave interaction problem.

Density (t = 2.5) Density (t = 5) WENO Flag

Hybrid-R

TS

Hybrid-M

R


2D Riemann problems (400× 400) :Hybrid Compact-RBF


Configuration5

Configuration12


2D Riemann problems (CPU Timings) :Hybrid Compact-RBF


Table IV: The CPU timings (in second) of the WENO-Z scheme and speedupfactors of the Hybrid schemes.

Case N×M WENO-Z Hybrid-MR Hybrid-RT Hybrid-RST

3400× 400 1922 2.08 1.58 1.88800× 800 17031 3.07 1.12 2.26

5400× 400 568 2.43 1.80 2.22800× 800 4674 3.05 1.07 2.33

12400× 400 628 2.42 1.91 2.31800× 800 4613 2.97 1.09 2.30


Summary and Further Work


Summary

We develop a hybrid compact-WENO finite different scheme withseveral high order such as RBF shocks detection algorithms.

Extend the high order shocks detection algorithms to unstructuredmesh and mesh free methods.

Investigate the multi-dimensional, multi-components, multi-physical,multi-scales problems in reactive Euler equations, the shallow waterequations, and the Naiver-Stokes equations using high order/resolutonHybrid scheme in structured and unstructured hybrid domains.


References


1. M.D. Buhmann, Radial Basis Functions: Theory and Implementations,Cambridge University Press, Cambridge, 2003.

2. J.-H. Jung, V. Durante, An iterative multiquadric radial basis function

method for the detection of local jump discontinuities, Appl. Numer.Math. 59 (2009), 1449–1446.

3. J.-H. Jung, S. Gottlied, S.O. Kim, Iterative adaptive RBF methods for

detection of edges in two-dimensional functions, Appl. Numer. Math. 61(2011) 77–91.

4. W. Trench, An Algorithm for the Inversion of Finite Toeplitz Matrices,SIAM J. Appl. Math. 12 (1964) 515–522.

5. J.W. Tukey, Exploratory data analysis, 1st ed., Behavioral Science:

Quantitative Methods, Addison-Wesley, (1997).

6. M.J. Vuik, J.K. Ryan, Automated parameters for troubled-cell indicators

using outlier de-tection, SIAM J. Sci. Comput.38 (2016), A84–A104.


Acknowledgement


Ocean University of China, ChinaProf. Gao Zhen

StudentsWen Xiao, Wang BaoShan, Wang Yinghua, Meng Xianjun

Beijing Institute of Technology, ChinaDr. Li Peng

Research funding supports were provided by National Natural Science Foundation of

China, Natural Science Foundation of Shandong Province, China Postdoctoral

Science Foundation, Fundamental Research Funds for the Central Universities, and

Startup fund from Ocean University of China (Don).


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