High Order Shock Detectors in Hybrid WENO-Compact Finite Difference Scheme for Hyperbolic Conservation Laws

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1 High Order Shock Detectors in Hybrid WENO-Compact Finite Difference Scheme for Hyperbolic Conservation Laws Advances in PDEs: Theory, Computation & Application to CFD ICERM, Brown University Providence, RI, USA Wai Sun, DON School of Mathematical Sciences Ocean University of China, Qingdao, Shandong, China August 20 - August 25, 2018

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

3 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 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 3 / 56

4 Saul Abarbanel Secondary frequencies in the wake of a circular cylinder with vortex shedding Journal of fluid mechanics, 1991 Saul S Abarbanel, Wai Sun Don, David Gottlieb, David H Rudy, James C Townsend The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Careful Study of the Boundary Condition SIAM JSC, 1993 Carpenter, Mark H ; Gottlieb, David ; Abarbanel, Saul ; Don, Wai-Sun Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 4 / 56

5 Saul Abarbanel Copyrighted by Disney. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 5 / 56

6 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 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 6 / 56

7 Nonlinear Hyperbolic PDE Finite Time Singularity Shock Q t + F(Q) = S. Burgers/Euler equations LINEAR scheme: The well-known Gibbs oscillations appears around the location of the shock (Fourier spectral method, Compact scheme ). NON-LINEAR Scheme: Numerical or physical based adaptive dissipation via upwinding or localized regularization(tvd, ENO, WENO, etc.) Q x Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 7 / 56

8 Sufficient Condition for Formal Order of WENO FD scheme The sufficient condition for the optimal order of the WENO finite difference scheme for the hyperbolic PDEs is ω ± k = ω± k d k = O( x r ), (1) It measures the deviation of the nonlinear ω k from the ideal weights d k. LARGE Lower order; Upwind; Larger Dissipation; Shock Capturing SMALL High order; Central; Lesser Dissipation; Higher Resolution A WELL-BALANCED definition of ω k is essential in capturing shocks essentially non-oscillatory; while resolving smooth solution. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 8 / 56

9 The Classical WENO-JS Scheme The lower order local smoothness indicators β k = x xi+ 1 2 x i 1 2 ( ) d 2 xi+ dx qk (x) dx+ x x i 1 2 ( ) d 2 2 dx 2qk (x) dx, (2) measures the sum of a normalized L 2 norm of the derivatives of the second degree polynomials q k (x) in substencil S k in cell x i = [x i 1,x 2 i+ 1]. 2 The nonlinear weights of the classical WENO-JS scheme (Jiang and Shu) are α k = d k (β k +ε) p, ω k = α k 2 l=0 α, l with a user defined power parameter p = 2 and the sensitivity parameter ε = 10 6,10 40, a fixed value, to avoid the division of zero when β k = 0. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 9 / 56

10 Smoothness Indicators In terms of cell averages values of f(x), f i, one has β 0 = (f i 2 2f i 1 +f i ) (f i 2 4f i 1 +3f i ) 2, β 1 = (f i 1 2f i +f i+1 ) (f i 1 f i+1 ) 2, β 2 = (f i 2f i+1 +f i+2 ) (3f i 4f i+1 +f i+2 ) 2. Their Taylor series expansions at x i are β 0 = f 2 i x 2 + ( f i 2 ) 3 f if i ( β 1 = f 2 13 i x f i + 1 ) 3 f if i ( β 2 = f 2 13 i x f i 2 ) 3 f if i x 4 ( 13 6 f i f i 1 ) 2 f if i x 5 +O( x 6 ), x 4 +O( x 6 ), x 4 + ( 13 6 f i f i 1 ) 2 f if i x 5 +O( x 6 ). Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 10 / 56

11 The Improved WENO-Z Scheme By introducing the optimal global smoothness indicator, ( ( ) ) 13 τ 5 = β 0 β f i f i f if i x 5 +O( x 6 ). One can define the nonlinear weights ( ( ) p ) αk z = d τ2r 1 k 1+ β k +ε, ω z k = α z k r 1 l=0 αz l. τ 5 measures the higher derivatives (smoothness) of f(x) on S 5, if existed. β k is of order O( x 2 ). τ 5 is of order O( x 5 ). τ 2r 1 β k is of order O( x r ), satisfying the sufficient condition δω k = O( x r ) for optimal order provided that f 0. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 11 / 56

12 Loss of Formal Order of WENO scheme at Critical Points Critical points of order n cp : f (0) =... = f (n cp) (0) = 0 and f (n cp+1) (0) 0). The nonlinear weights of the WENO-JS and WENO-Z schemes ω k do not satisfy the sufficient conditions in the present of critical points of order (n cp 1) for WENO-JS and (n cp 2) for WENO-Z, for a fixed small ε. x WENO-JS7 WENO-Z7 (p = 2) WENO-Z7 (p = 3) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table I: Rate of convergence at a second order critical point (n cp = 2) for the seventh order (r = 4) WENO scheme with ε = Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 12 / 56

13 Numerical Dissipation of WENO Scheme Entropy with N = 1500 at time t = Reference FC-WENO-Z5 WENO-Z5 Reference FC-WENO-Z5 WENO-Z Entropy 0.45 Entropy x x Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 13 / 56

14 Disadvantages of using WENO Scheme The WENO-JS/Z schemes, in order to guarantee essentially non-oscillatory at HIGH 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 to figure out the right version of WENO scheme to use and parameters to tune for a given scheme and problem. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 14 / 56

15 Space-Time Adaptive Hybrid scheme A natural remedy to alleviate some of these difficulties is to AVOID using WENO NONLINEAR scheme at the known SMOOTH regions of the solution at a given time. Hybridization of the high order WENO nonlinear scheme for discontinuous substencils (subdomains) and varieties of high order linear schemes for smooth substencils (subdomains) have been devised. Example: Hybrid Compact-WENO scheme (Pirozzoli, Ren, etc.) Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 15 / 56

16 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 and efficient.) 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 requires boundary closure.) Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 16 / 56

17 Hybrid Scheme High Order linear spectral schemes can be Chebyshev collocation method in multi-domain framework : (global, relatively simple, non-dissipative, non-dispersive, spectral accuracy, FFT, non-uniform grid, require interpolation between subdomains 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 between subdomains, solution of a severe to mild ill-condition system and a careful choice of user defined parameters.) Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 17 / 56

18 High Order Compact scheme A 6th-order (c r = 6) compact finite difference scheme on a uniformly spaced grids, with c = 1 36 x, Ag = cbg+b, A = 1 1/3 1/3 1 1/ /3 1 1/3 1/3 1,B = b = c( g 1 28g 0, g 0,0,,0,g N,28g N +g N+1) T 1 3 (g 0,0,0,,0,0,g N) T, where g 1 = g(x 0 x) and g N+1 = g(x N + x) are the ghost points. g 0 and g N are computed by the WENO-Z scheme. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 18 / 56

19 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 x i that all the grid points inside the buffer zone are flagged as non-smooth stencils. This condition prevents the computation of the derivative of the fluxes by the 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 19 / 56

20 High Order Smoothness analysis A successful implementation of a high order Hybrid scheme strongly depends on the ability to obtain HIGH ORDER accurate information on the smoothness of and sharp detection of high gradients/shocks/edges in the solution 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 grid with 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 edge detection analysis. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 20 / 56

21 Multi-resolution Analysis (Harten) Given an initial number of the grid points N 0 and grid spacing x 0, we shall consider a set of nested dyadic grids up to level L < log 2 N 0, G k = {x k i, i = 0,...,N k }, 0 k L, (3) where x k i = i x k with x k = 2 k x 0, N k = 2 k N 0 and the cell averages of function u at x k i : ū k i = 1 x k x k i x k i 1 u(x)dx. (4) Define the 2s = q 1 degree POLYNOMIAL ũ k 2i 1 (5) that approximates ū k 2i 1 interpolating ūk i+l, l s at xk i+l. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 21 / 56

22 Multi-resolution analysis (Cont.) The approximation error (or multi-resolution coefficients), d i = ū 0 2i 1 ũ 0 2i 1, (6) at x i, has the property that if u(x) is a C p 1 function, then d i { (p) [u i ] x p p q u (q) i x q 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 22 / 56

23 Multi-resolution analysis (Cont.) MR 3rd Order MR 8th Order f(x) f(x) 3rd Order 5th Order 7th Order 5 4 MR Coefficient d f(x) MR Coefficient d(x) f(x) x x 0 Figure 1: (Left) The third and eighth order MR coefficients d i of a piecewise analytic function. (Right) The third, fifth and seventh order MR coefficients d i of the density ρ(x) of the Mach 3 Shock-Entropy wave interaction problem. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 23 / 56

24 2D Detonation Diffraction Problem : Hybrid-Compact-MR Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 24 / 56

25 2D Detonation Diffraction Problem : Hybrid-Compact-MR The MR flags in the x and y-directions of detonation diffraction around a 90 o corner with N x N y = WENO Flag in x WENO Flag in y Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 25 / 56

26 2D Detonation Diffraction Problem : Hybrid-Compact-MR Table II: Comparative CPU timing and speedup for the two-dimensional detonation diffraction problem. 2r 1 c r N x N y WENO-Z Hybrid-Compact Speedup Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 26 / 56

27 Multi-resolution analysis (Cont.) Pros Cons 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. Require a parameter ǫ MR (typical 10 3 ) to determine where the high gradients/shocks are. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 27 / 56

28 Conjugate Fourier Analysis Consider the power series on the unit circle with z = exp(ix), F(z) = 1 2 a 0 + (a k ib k )z k, (8) k=1 with the Fourier coefficients a k and b k. Its REAL part is the infinite Fourier series of a periodical function f(x) : f(x) = 1 2 a 0 + (a k coskx+b k sinkx), (9) k=1 Its IMAGINARY part is called the conjugate Fourier series (Zygmund (1968), Lukács (1920), Fejér (1913) etc.): f(x) = (a k sinkx b k coskx), (10) k=1 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 28 / 56

29 Conjugate Fourier Analysis (Cont.) Truncate the series into N degree trigonometric polynomials : f N (x) = 1 N 2 a 0 + (a k coskx+b k sinkx), (11) f N (x) = ( f N ) (x) = f N (x) = k=1 N (a k sinkx b k coskx). (12) k=1 N k(a k coskx+b k sinkx). (13) k=1 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 29 / 56

30 Conjugate Fourier Analysis (Cont.) f(x) fn (x) fn (x) f(x) FC x x x Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 30 / 56

31 Conjugate Fourier Analysis (Cont.) The concentration property of f N (x) gives the location and jump of a discontinuity based on the following Lemma 1 : Lemma 1. Let f(x) be a 2π-periodic piecewise smooth function, except a single jump discontinuity at x = ξ with an associated jump then [f](ξ) := f(ξ + ) f(ξ ), (14) π logn f N (x) [f](x)δ ξ (x) = { [f](ξ), x = ξ, 0, otherwise. (15) where δ ξ is the Dirac distribution located at ξ. 1 Gelb and Tadmor, 97, 99 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 31 / 56

32 Conjugate Fourier Analysis (Cont.) Pros Cons 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). More complex to implement. Slower than MR analysis with a O(5N logn) floating points operation. ) The rate of convergence is only O( 1 logn at best. Require ǫ cf to identify the local of the discontinuities. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 32 / 56

33 Iterative Conjugate Fourier Shock Detection Algorithm (Right) Iterative cf shock detector with N = 128. (Left) Iterative cf shock detector with N = 256. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 33 / 56

34 Two dimensional Riemann IVP problem (Hybrid Compact-cF) Rho Pressure Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 34 / 56

35 Two dimensional Riemann IVP problem (Hybrid Compact-cF) WENO Flag in x WENO Flag in y Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 35 / 56

36 Multi-Quadric Radial Basis Function (MQ-RBF) methods φ i (x) = (x x i ) 2 +ε i2, x i 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 λ i φ i (x), g (x) = i=1 n λ j (x x j )/φ i (x). (16) j=1 where λ i are the RBF expansion coefficients. By defining the interpolation matrix M with M ij = (x i x j ) 2 +ε j2, one has λ = M 1 f, (17) By defining the differentiation matrix D with D ij = (x i x j )/M ij, one has g = Dλ, (18) Remark : M is a symmetric Toeplitz matrix on a uniformly spaced mesh. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 36 / 56

37 Concentration Map and Edge Set A normalized concentration map C, that reflects the smoothness of the function and indicates the possible locations of the local jumps, is defined by { Ĉ = Ĉ i Ĉi = C } i maxc,c i = Dλ 2, x i X, i = 1,2,,n. (19) The edge set S can then be defined S = {S i S i = x i, Ĉi η,g (x i ) 0,x i X,i = 1,2,,n}, (20) where η (0,1) is the given tolerance parameter. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 37 / 56

38 1D edge detection with IAMQ-RBF method A simple sign function is defined as { 1, 1 x < 0, f(x) = 1, 0 x 1. (21) Figure 2: Left : Edges detected by the IAMQ-RBF-Fast method with ǫ = 0.1,η = 0.5 and n = 50. Right: The normalized concentration map Ĉ. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 38 / 56

39 Iterative Adaptive Multi-Quadric Radial Basis Function (IAMQ-RBF) methods The iterative adaptive MQ-RBF (IAMQ-RBF) method is based on the growth/decay properties of the coefficients λ (Jung et al. 09). That is, λ grows (decay) exponentially near and at (away from) the discontinuities. The IAMQ-RBF method is iteratively 1. Set the constant shaper parameter {ε i = ǫ} to find the edge set S from the concentration map Ĉ. 2. Iteratively/Adaptively set ǫ i = 0 for those x i S to remove the detection of the jumps at x i in the subsequent iterations. 3. STOP when the residual λ k+1 λ k 2 δ,(δ = 10 8 ). Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 39 / 56

40 Fast solver for Toeplitz system in RBF method Problem : Step 2 requires to find the time consuming inverse of a perturbed NON-SYMMETRIC Toeplitz-ǫ matrix system of the form Mx = (T+P)x = b at each iteration. ε h 2... (nh)2 +ε 2. M = h2 +ε h2 +ε 2 (nh)2 +ε 2 ((n 1)h) 2 h2 +ε 2 ε Solution : Find a FAST algorithm! Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 40 / 56

41 Fast solver for Toeplitz system in RBF method In the iterative steps of the IAMQ-RBF method, the perturbed Toeplitz-ǫ matrices are generated and one need an efficient solver for the system (T+P)x = b, where T R n n, and P R n n is a zero matrix except its columns n 1,n 2,,n m and m n. Write P = UV T where U R n m with the columns being the non-zero columns of P, and V R n m is the permutation matrix, that is, V = (e n1,e n2,,e nm ) R n m, with e i R 1 n being the i-th unit column vector. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 41 / 56

42 The solver for Toeplitz-ǫ matrix By the Sherman-Morrison-Woodbury formula, one has (T+UV T ) 1 = T 1 T 1 U(I m +V T T 1 U) 1 V T T 1, (22) where I m R m m is an identity matrix. Thus, we only need to solve two Toeplitz systems { TQ = U, (23) Tv = b, and a small (m m) system Bw = V T v, (24) where to obtain the solution B = I m +V T Q R m m. (25) x = v Qw. (26) Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 42 / 56

43 Fast recursive O(n 2 ) algorithm for T 1 NEED a Fast O(n 2 ) direct solver for T 1 Instead of the (O(n 3 ) classical Gaussian-Elimination method, we take advantage of the special structure of the Toeplitz matrix and to compute its inverse recursively including the O(n 2 ) Levinson-Durbin algorithm (Trench 64), which also employed the Yule-Walker algorithm. Both of them are O(n 2 ) recursion algorithms. (See reference for details) We shall refer the algorithm for solving (T+P)x = b as IAMQ-RBF-Fast method. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 43 / 56

44 1D edge detection with IAMQ-RBF-Fast method 15 f(x) L1PA MR IAMQ-RBF IAMQ-RBF-Fast 10 f(x) x 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 44 / 56

45 1D edge detection with IAMQ-RBF-Fast method First Second Third Fourth Flag Flag Flag Flag x x x x C C C C x x x x Figure 4: The edges of the piecewise linear function detected by the IAMQ- RBF-Fast method at each iteration. Bottom: The normalized concentration map C at each iteration. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 45 / 56

46 2D edge detection with IAMQ-RBF-Fast method The final detected edges of Shepp-Logan image, sunflower image and three classical photos downloaded from USC Signal and Image Processing Institute Data base with the size of 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 46 / 56

47 2D edge detection with IAMQ-RBF-Fast method Figure 6: Final edges of the sunflower image with the size of Left: The original image. Right: The detected edges by the IAMQ-RBF-Fast method with ǫ = 0.1,η = 0.4. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 47 / 56

48 2D edge detection with IAMQ-RBF-Fast method Figure 7: Final edges of the clock image with the size of Left: The original image. Right: The detected edges by the IAMQ-RBF-Fast method with ǫ = 0.1,η = Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 48 / 56

49 2D edge detection with IAMQ-RBF-Fast method The comparison with the original method on CPU timing shown in Table III demonstrates that the present method is an effective (at least three times faster 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 Clock image Airplane image Shepp-Logan image Resolution test image Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 49 / 56

50 Edge Detection with IAMQ-RBF-Fast method Pros Fast with the fast O(n 2 ) direct solver. Sharp detection of the location of discontinuities. Fairly straightforward to implement. Captures almost ALL discontinuities within a finite number of iterations. 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 50 / 56

51 Examples: 1D Shock Detection Consider the extended Mach 3 shock density wave interaction problem. Density (t = 2.5) Density (t = 5) WENO Flag Hybrid-MR Hybrid-RTS Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 51 / 56

52 2D Riemann problems ( ) : Hybrid Compact-RBF Configuration 12 Configuration 5 Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 52 / 56

53 2D Riemann problems (CPU Timings) : Hybrid Compact-RBF Table IV: The CPU timings (in second) of the WENO-Z scheme and speedup factors of the Hybrid schemes. Case N M WENO-Z Hybrid-MR Hybrid-RT Hybrid-RST Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 53 / 56

54 Summary and Further Work Summary We develop a hybrid compact-weno finite different scheme with several high order such as RBF shocks detection algorithms. Extend the high order shocks detection algorithms to unstructured mesh and mesh free methods. Investigate the multi-dimensional, multi-components, multi-physical, multi-scales problems in reactive Euler equations, the shallow water equations, and the Naiver-Stokes equations using high order/resoluton Hybrid scheme in structured and unstructured hybrid domains. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 54 / 56

55 References 1. M.D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, J.-H. Jung, V. Durante, An iterative multiquadric radial basis function method for the detection of local jump discontinuities, Appl. Numer. Math. 59 (2009), 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) W. Trench, An Algorithm for the Inversion of Finite Toeplitz Matrices, SIAM J. Appl. Math. 12 (1964) 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. Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 55 / 56

56 Acknowledgement Ocean University of China, China Prof. Gao Zhen Students Wen Xiao, Wang BaoShan, Wang Yinghua, Meng Xianjun Beijing Institute of Technology, China Dr. 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). Ocean University of China High Order Shock Detectors in Hybrid WENO-Compact Scheme 56 / 56