Two-waves PVM-WAF method for non-conservative systems

Transcription

1 Two-waves PVM-WAF method for non-conservative systems Manuel J. Castro Díaz 1, E.D Fernández Nieto, Gladys Narbona Reina and Marc de la Asunción 1 1 Departamento de Análisis Matemático University of Málaga (Spain) Departmento de Matemática Aplicada University of Sevilla (Spain) Hyp 01, Padova 5-9 June 01

2 Outline 1 Introduction Path-conservative Roe-based schemes PVM methods WAF schemes PVMU-WAF method 3 Numerical tests 4 Conclusions

3 Model problem Let us consider the system w t + F(w) x + B(w) w x = G(w)H x, (1) where w(x, t) takes values on an open convex set O R N, F is a regular function from O to R N, B is a regular matrix function from O to M N N(R), G is a function from O to R N, and H is a function from R to R. By adding to (1) the equation H t = 0, the system (1) can be rewritten under the form where W is the augmented vector and W = W t + A(W) W x = 0, () [ w H ] Ω = O R R N+1

4 Model problem Let us consider the system w t + F(w) x + B(w) w x = G(w)H x, (1) where w(x, t) takes values on an open convex set O R N, F is a regular function from O to R N, B is a regular matrix function from O to M N N(R), G is a function from O to R N, and H is a function from R to R. By adding to (1) the equation H t = 0, the system (1) can be rewritten under the form where A(W) is the matrix whose block structure is given by: [ A(w) G(w) A(W) = 0 0 W t + A(W) W x = 0, () ], where A(w) = J(w) + B(w), being J(w) = F w (w).

5 Difficulties Main difficulties Non conservative products A(W) W x. Solutions may develop discontinuities and the concept of weak solution in the sense of distributions cannot be used. The theory introduced by DLM 1995 is used here to define the weak solutions of the system. This theory allows one to give a sense to the non conservative terms of the system as Borel measures provided a prescribed family of paths in the space of states. Derivation of numerical schemes for non-conservative systems: path-conservative numerical schemes (Parés 006). The eigenstructure of systems like bilayer Shallow-Water system or two-phase flow model of Pitman Le are not explicitly known: PVM and/or WAF schemes.

6 PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: where w n+1 i = w n i t ( D + i 1/ x + i+1/) D, (3) x and t are, for simplicity, assumed to be constant; w n i is the approximation provided by the numerical scheme of the cell average of the exact solution at the i-th cell, I i = [x i 1/, x i+1/ ] at the n-th time level t n = n t. H i is the cell average of the function H(x).

7 PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: w n+1 i = w n i t ( D + i 1/ x + i+1/) D, (3) D ± i+1/ = 1 ( F(wi+1) F(w i) + B i+1/ (w i+1 w i) G i+1/ (H i+1 H i) ) ± Q i+1/ (w i+1 w i A 1 i+1/ G i+1/(h i+1 H i)), (4) where A i+1/ = J i+1/ + B i+1/. Here, J i+1/ is a Roe matrix of the Jacobian of the flux F in the usual sense: B i+1/ (w i+1 w i) = J i+1/ (w i+1 w i) = F(w i+1) F(w i); B(Φ w(s; W i, W i+1)) Φw (s; Wi, Wi+1) ds; s G i+1/ (H i+1 H i) = G(Φ w(s; W i, W i+1)) ΦH (s; Wi, Wi+1) ds; 0 s Q i+1/ is a numerical viscosity matrix.

8 PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: w n+1 i = w n i t ( D + i 1/ x + i+1/) D, (3) D ± i+1/ = 1 ( F(wi+1) F(w i) + B i+1/ (w i+1 w i) G i+1/ (H i+1 H i) ) ± Q i+1/ (w i+1 w i A 1 i+1/ G i+1/(h i+1 H i)), (4) Conservative systems If the system is conservative and F i+1/ = F(wi) + F(wi+1) 1 Q i+1/(w i+1 w i) is a conservative flux, where Q i+1/ is defined in terms of J i+1/, then D i+1/ = F i+1/ F(w i) D + i+1/ = F(wi+1) F i+1/.

9 PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: w n+1 i = w n i t ( D + i 1/ x + i+1/) D, (3) D ± i+1/ = 1 ( F(wi+1) F(w i) + B i+1/ (w i+1 w i) G i+1/ (H i+1 H i) ) ± Q i+1/ (w i+1 w i A 1 i+1/ G i+1/(h i+1 H i)), (4) Different numerical schemes can be obtained for different definitions of Q i+1/

10 PC-Roe-based schemes II Roe scheme corresponds to the choice Q i+1/ = A i+1/, Lax-Friedrichs scheme: being Id the identity matrix. Lax-Wendroff scheme: Q i+1/ = x t Id, Q i+1/ = t x A i+1/, FORCE and GFORCE schemes: Q i+1/ = (1 ω) x t Id + ω t x A i+1/, with ω = 0.5 and ω = 1, respectively, being α the CFL parameter. 1+α

11 PVM methods We propose a class of finite volume methods defined by being P l (x) a polinomial of degree l, Q i+1/ = P l (A i+1/ ), P l (x) = l j=0 α i+1/ j x j, and A i+1/ a Roe matrix. That is, Q i+1/ can be seen as a Polynomial Viscosity Matrix (PVM). See also: P. Degond, P.F. Peyrard, G. Russo, Ph. Villedieu. Polynomial upwind schemes for hyperbolic systems. C. R. Acad. Sci. Paris 1 38, , 1999.

12 PVM methods We propose a class of finite volume methods defined by being P l (x) a polinomial of degree l, Q i+1/ = P l (A i+1/ ), P l (x) = l j=0 α i+1/ j x j, and A i+1/ a Roe matrix. That is, Q i+1/ can be seen as a Polynomial Viscosity Matrix (PVM). Q i+1/ has the same eigenvectors than A i+1/ and if λ i+1/ is an eigenvalue of A i+1/, then P l (λ i+1/ ) is an eigenvalue of Q i+1/.

13 PVM methods We propose a class of finite volume methods defined by being P l (x) a polinomial of degree l, Q i+1/ = P l (A i+1/ ), P l (x) = l j=0 α i+1/ j x j, and A i+1/ a Roe matrix. That is, Q i+1/ can be seen as a Polynomial Viscosity Matrix (PVM). Some well-known solvers as Lax-Friedrichs, Rusanov, FORCE/GFORCE, HLL, Roe, Lax-Wendroff,... can be recovered as PVM methods

14 PVM-1U(S L, S R ) or HLL method P 1(x) = α 0 + α 1 x such as P 1(S L) = S L, P 1(S R) = S R. Q i+1/ = α 0Id + α 1A i+1/ PVM 1U(S L,S R ) S L λ 1 λ λ j... λ N S R

15 PVM-1U(S L, S R ) or HLL method The usual HLL scheme coincides with PVM-1U(S L, S R) in the case of conservative systems. Let us suppose that the system is conservative. Then, the conservative flux associated to PVM-1U(S L,S R) is F i+1/ = D i+1/ + F(wi). Taking into account that then F i+1/ = α 0 = SR SL SL SR SR SL, α 1 =, S R S L S R S L F(wi)(SR + SR SL SL ) + F(wi+1)(SR SR SL + SL ) S R S L (SR SL SL SR )(wi+1 wi) S R S L = S+ R F(wi) S L F(wi+1) + (S+ R S L )(wi+1 wi) S + R S L which is a compact definition of the HLL flux, being S + R = min(sl, 0). S L = max(sr, 0) and

16 PVM-U(S L, S R ) method such as where P (x) = α 0 + α 1x + α x, P (S m) = S m, P (S M) = S M, P (S M) = sgn(s M), { SL if S L S R, S M = if S L < S R. S R { SR if S L S R, S m = if S L < S R. S L PVM U(S L,S R ) SL λ1 λ λj... λn SR

17 WAF method Let us consider the following Riemann problem: w + F(w) = 0; t x w(x, 0) = { wi x < 0; w i+1 x > 0. We denote by S i for i = 1,, N some approximation of the characteristic velocities. S 1 t S! t "! t / " 1 " 3 0 #! x /! x / x F WAF i+1/ = 1 x x/ x/ F(w(x, t ))dx.

18 where F (k) i+1/ is the value of the flux function in the interval k, WAF method (Toro [1989]) S 1 t S! t "! t / " 1 " 3 0 #! x /! x / x ω k = 1 (c k c k 1 ), c 0 = 1, c N+1 = 1 and c l = t x S l, for 1 l N, N is the number of waves. So, we can write the numerical flux as follows: F WAF i+1/ = 1 (Fi + Fi+1) 1 N k=1 c k F (k) i+1/,

19 WAF method TVD WAF method (Toro [1989]) We denote χ(v) a flux limiter function, and Ψ(v, c) = 1 (1 c )χ(v). So the TVD-WAF flux function is defined by: where F WAF i+1/ = 1 (Fi + Fi+1) 1 N k=1 sign(s k )Ψ k F (k) i+1/, Ψ k = Ψ(v (k), c k ) = 1 (1 c k )χ(v (k) ). Some suitable choices for χ can be found in [Toro]. Let us consider here the Van Albada s limiter: p (k) 0 if v (k) i p (k) i 1 0 χ(v (k) ) = v (k) (1 + v (k) ) if v (k) 0, where v (k) p (k) i+1 = p(k) i p (k) 1 + v (k) i+ p(k) i+1 p (k) i+1 p(k) i being p (k) a scalar value. if S k > 0 if S k < 0,

20 HLL-WAF method We consider now N =, S 1 = S L, S = S R, F 1 i+1/ = F(w i), F 3 i+1/ = F(wi+1) and F () SRFi SLFi+1 + SRSL(wi+1 wi) i+1/ = S R S L F HLL-WAF i+1/ = 1 (Fi + Fi+1) 1 (ν1(χl, χr)(wi+1 wi) + ν(χl, χr)(fi+1 Fi)) 1 t (µ1(χl, χr)(wi+1 wi) + µ(χl, χr)(fi+1 Fi)), x where ν 1(χ L, χ R) = ν (χ L, χ R) = µ 1(χ L, χ R) = µ (χ L, χ R) = S Rχ R S Lχ L S R S L. SLSR((1 χl)sgn(sl) (1 χr)sgn(sr)) S R S L (1 χr) SR (1 χl) SL S R S L SLSR(SLχL SRχR) S R S L

21 HLL-WAF method as a PVM-type method Then, we can rewrite the HLL-WAF method as follows: F HLL-WAF i+1/ = 1 (Fi+1 + Fi) 1 QHLL WAF i+1/ (w i+1 w i), with Q HLL WAF i+1/ (χ L, χ R) = Q HLL WAF t o1,i+1/ (χl, χr) + x QHLL WAF o,i+1/ (χl, χr) Q HLL WAF o1,i+1/ (χl, χr) = ν1(χl, χr)i + ν(χl, χr)a i+1/ Q HLL WAF o,i+1/ (χl, χr) = µ1(χl, χr)i + µ(χl, χr)a i+1/. That is, the usual two-waves HLL-WAF method can be seen as a non-linear combination of two PVM schemes associated to the first order polynomials: P o1 1 (x) = ν 1(χ L, χ R) + ν (χ L, χ R)x and P o 1 (x) = µ 1(χ L, χ R) + µ (χ L, χ R)x.

22 HLL-WAF method as a PVM-type method y=p 1 HLL (x) y=p 1 LW (x) S1 λ1 λ λj... λn S

23 HLL-WAF method as a PVM-type method y=p 1 HLL (x) y=p 1 LW (x) S1 λ1 λ λj... λn S Remarks For systems with N=. If S 1 = λ 1,i+1/ and S = λ,i+1/ being λ j,i+1/ the eigenvalues of Roe matrix, then HLL-WAF method coincides with Lax-Wendroff method if (χ L = χ R = 1). For N > it is not true! HLL-WAF method coincides with HLL if (χ L = χ R = 0) (N ).

24 HLL-WAF method as a PVM-type method y=p 1 HLL (x) y=p 1 LW (x) S1 λ1 λ λj... λn S Objective We want to define a new two-wave WAF method so that coincides with Lax-Wendroff for N > if χ L = χ R = 1

25 Two-waves PVMU-WAF methods We consider F U-FL Fi + Fi+1 i+1/ = where Q U FL i+1/ (χ L, χ R ) is defined as follows: with 1 QU FL i+1/ (χl, χr)(wi+1 wi), Q U FL i+1/ (χ L, χ R ) = Q U FL o1,i+1/ (χ L, χ R ) + t x QU FL o,i+1/ (χ L, χ R ), Q U FL o1,i+1/ (χ L, χ R ) = sgn(s L)(1 χ L ) + sgn(s R )(1 χ R ) A i+1/ and + sgn(s R)(1 χ R ) P,αR (A i+1/ ) sgn(s L)(1 χ L ) P,αL (A i+1/ ), Q U FL o,i+1/ (χ L, χ R ) = S Lχ L + S R χ R A i+1/ + S Rχ R P,αR (A i+1/ ) S Lχ L P,α L (A i+1/ ), where α K = 1 (1 χ K )(1 α), K = L, R, α = (S R S L )sgn(s M ) (S R + S L ), 4S M (S L + S R )

26 Two-waves PVMU-WAF methods We consider F U-FL Fi + Fi+1 i+1/ = where Q U FL i+1/ (χ L, χ R ) is defined as follows: with 1 QU FL i+1/ (χl, χr)(wi+1 wi), Q U FL i+1/ (χ L, χ R ) = Q U FL o1,i+1/ (χ L, χ R ) + t x QU FL o,i+1/ (χ L, χ R ), Q U FL o1,i+1/ (χ L, χ R ) = sgn(s L)(1 χ L ) + sgn(s R )(1 χ R ) A i+1/ and + sgn(s R)(1 χ R ) P,αR (A i+1/ ) sgn(s L)(1 χ L ) P,αL (A i+1/ ), Q U FL o,i+1/ (χ L, χ R ) = S Lχ L + S R χ R A i+1/ + S Rχ R P,αR (A i+1/ ) S Lχ L P,α L (A i+1/ ), and P,α (x) = αp M (x) + (1 α)p 1M (x), P 1M (x) = S RS L S R S L + S R + S L S R S L x.

27 Two-wave PVM-U-WAF method Properties Only uses the information of the two fastest waves. If N = it coindices with the usual HLL-WAF scheme. If χ L = χ R = 0, then we recover the PVM-U first order scheme for N. If χ L = χ R = 1, then scheme reduces to Lax-Wendroff scheme for N.

28 Two-wave PVM-U-WAF method Remark A natural extension to balance laws and non-conservative system is straightforward: with w n+1 i = w n i t ( D + i 1/ x + i+1/) D, D ± i+1/ = 1 ( F(wi+1) F(w i) + B i+1/ (w i+1 w i) G i+1/ (H i+1 H i) ) ± Q i+1/ (w i+1 w i A 1 i+1/ G i+1/(h i+1 H i)), being But it is not second order. Q i+1/ = Q U FL i+1/ (χl, χr)

29 Two-wave PVM-U-WAF method To recover the second order, new terms appear in the Lax-Wendroff scheme due to the non-conservative products and source terms (see Castro, Pares & Toro Math. Comp. 010): w n+1 i = w n i t ( D + i 1/ x + ) t D i+1/ + 4 x (R(χL, χr)n i 1/ + R(χ L, χ R) n i+1/) with R(χ L, χ R ) n i+1/ = 1 ( χ L DA(W i )[A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i ), w i+1 w i ] + χ R DA(W i+1 )[A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i ), w i+1 w i ] χ L DA(W i )[w i+1 w i, A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i )] χ R DA(W i+1 )[w i+1 w i, A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i )] χ L G w(w i )(A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i ))(H i+1 H i ) ) χ R G w(w i+1 )(A i+1/ (w i+1 w i ) G i+1/ (H i+1 H i ))(H i+1 H i ) ( ) N being DA(W)[U, V] = l=1 u l wl A(W) V and wl A(W) is the N N matrix whose (i, j) element is wl a ij (W). G w(w) denotes the Jacobian matrix of G(w).

30 Multilayer shallow water system th j + xq j = 0, tq j + x( q j + 1 h j gh j ) + gh j x(z b + h k + ρ k h k ) = 0. ρ j k>j k<j j = 1,..., m, where m is the number of layers, h j, j = 1,..., m are the fluid depths, q j = h ju j are the discharges, u j are the velocites and z b (x) is the topography. g is the gravity constant and ρ j the densisites of the stratified fluid layers, with 0 < ρ 1 < < ρ m. We use the Van Albada s limiter with a smooth indicator of the fluid interfaces. Total energy of the system provides also good results.

31 1LSW: Stationary subcritical solution over bump I = [0, 0]. z b (x) = 0.e 0.16(x 10) Boundary conditions: q(0, t) = 4.4 and h(0, t) =.0 x = 1/0. cfl = 0.9. Nodes L 1 err h L 1 order h L 1 err q L 1 order q Table: Errors and order. Subcritical stationary solution.

32 1LSW: Order of accuracy I = [0, 1], T = 0.1. z b (x) = sen (πx). Initial condition: h(x, 0) = 5 + e cos(πx), q(x, 0) = sin(cos(πx)), CFL = 0.8. Reference solution computed with ROE scheme with x = 1/1800. Nodes L 1 err h L 1 order h L 1 err q L 1 order q Table: Errors and order.

33 1LSW stationary transcritical flow with a shock I = [0, 0] bottom topography: { (x 10) 8 < x < 1 z b (x) = 0 otherwise. Initial condition h(x, 0) = 0.33 z b, q(x, 0) = 0. Boundary conditions: q = 0.18 at x=0 and h = 0.33 at x = 0 x = 1/10 and CFL = 0.9. Concerning CPU time similar results are obtained for Roe solver and PVM-U WAF method.

34 1LSW stationary transcritical flow with a shock Exact solution 0.1 Roe PVM U WAF 0.05 HLL WAF PVM U Bottom (a) Free surface an bottom topography

35 LSW: Internal dam break problem. I = [0, 10]. zb(x) = 0 Initial condition: q 1(x, 0) = q { (x, 0) = 0, x [0, 10], 0.9 if x < 5, h 1(x, 0) = 0.1 if x 5, h (x, 0) = 1.0 h 1(x, 0) x [0, 10]. Open boundary conditions x = 1/0. A reference solution is computed Roe scheme with x = 1/00. r = 0.99, cfl = 0.9. Concerning CPU time PVM-U-WAF method is.8 times faster than Roe and similar to original HLL-WAF scheme.

36 LSW: Internal dam break problem Reference solution Roe PVM U WAF (b) PVM-U-WAF and Roe Reference solution HLL WAF PVM U (c) HLL-WAF and PVM-U Figure: Internal dam break: free surface and interface at t = 0 seg.

37 LSW: Stationary transcritical flow with an internal shock I = [0, 10] Bottom topography: z b (x) = 0.5e (x 5). Initial condition: q 1(x, 0) = q (x, 0) = 0. and h 1(x, 0) = { 0.48 if x < 5, 0.5 if x 5, h (x, 0) = 1 h 1(x, 0) z b (x), ρ 1/ρ = Free boundary conditions. CFL = 0.9, x = 1/0. Reference solution computed with x = 1/00.

38 LSW: Stationary transcritical flow with an internal shock Reference solution 0. Roe PVM U WAF HLL WAF PVM U bottom (a) Free surface, bottom topography and interface

39 4LSW: Internal dam breaks. I = [0, 10] Bottom topography: z b (x) = 0.0 Initial condition: q i(x, 0) = 0., i = 1,..., 4 and { 0.9 if x < 5, h 1(x, 0) = 0.1 if x 5, h (x, 0) = 1 h 1(x, 0), h 3(x, 0) = h 1(x, 0), h 4(x, 0) = h (x, 0) ρ 1/ρ 4 = 0.85, ρ /ρ 4 = 0.9, ρ 3/ρ 4 = Free boundary conditions. CFL = 0.9, x = 1/0. Reference solution computed with x = 1/00. Concerning CPU time PVM-U-WAF method is 9.8 times faster than Roe and similar to original HLL-WAF scheme.

40 4LSW: Internal dam breaks Reference solution Roe 0. Reference solution HLL WAF PVM U WAF PVM U (b) PVM-U-WAF and Roe (c) HLL-WAF and PVM-U Figure: Internal dam breaks: free surface and interfaces at t = 5 seg.

41 D 1LSW: Circular dam break D = [, ] [, ] Bottom topography: zb(x, y) = 0.8 e x y Initial condition: q x(x, y, 0) = q y(x, y, 0) = 0 and { 1 zb(x, y) if x + y h(x, y, 0) = < zb(x, y) otherwise Wall boundary conditions x = y = 1/100, CFL = 0.9. Three implementations are considered: first and second order HLL and PVM-U-WAF method. Algorithms implemented on GPUs: speedups of more than 00 for the three numerical schemes, The extension by the method of lines of 1D PVM-U-WAF method to multidimensional problems is NOT second order accurate.

42 D 1LSW: Circular dam break (a) PVM-U WAF t = 1.0 s (b) HLL t = 1.0 s Figure: D circular dam break: free surface at t = 1 seg.

43 D 1LSW: Circular dam break (a) PVM-U WAF t = 1.0 s (b) Second order HLL t = 1.0 s Figure: D circular dam break: free surface at t = 1 seg.

44 D 1LSW: Circular dam break (a) PVM-U WAF t =.0 s (b) HLL t =.0 s Figure: D circular dam break: free surface at t = 1 seg.

45 D 1LSW: Circular dam break (a) PVM-U WAF t =.0 s (b) Second order HLL t =.0 s Figure: D circular dam break: free surface at t = seg.

46 Conclusions Conclusions The original two-wave HLL-WAF method can be seen as a PVM-based flux-limiting scheme. A new two-wave WAF method that ensured second order of accuracy for N > is defined using PVM framework. It can be applied to conservative, balance laws and non-conservative systems. Its performance increases with the complexity of the system. It can be 10 times faster than Roe solver for the 1D 4LSW. Extension to D that preserves second order accuracy: comming soon, it is NOT straight forward.