Grid adaptivity for systems of conservation laws

Transcription

1 Grid adaptivity for systems of conservation laws M. Semplice 1 G. Puppo 2 1 Dipartimento di Matematica Università di Torino 2 Dipartimento di Scienze Matematiche Politecnico di Torino Numerical Aspects of Hyperbolic Balance Laws and Related Problems Ferrara, Apr 2012 M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

2 A recipe for an adaptive scheme Ingredients of a semidiscrete scheme Recipe nonuniform (finite volume) grid refine/coarsen strategy (and indicator) time marching method reconstructions and numerical fluxes Mix together all ingredients, stir until it compile and runs smoothly and enjoy the movie show! plain C: 1D, second order, local timestepping C++, using dune 1 library for grid management: dimension independent, up to 3rd order (not yet!) 1 Distributed and Unified Numerics Environment M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

3 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions Puppo G., M.S. Numerical entropy and adaptivity for finite volume schemes Comm. in Comput. Phys., Puppo G., M.S. Adaptive grids and the entropy error indicator Proceedings of HYP2010, to appear. M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

4 Locally refined grids One grid versus many subgrids one grid + refined subgrids (rectangular patches) exploits fast algorithms on logically cartesian meshes need care to be conservative only one grid no need to be cartesian, no need to use squares attach data only to active cells M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

5 2D grids and quad-trees Locally refined grids Data are attached only to coloured cells M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

6 Locally refined grids One dimensional grids and binary trees Locally refine cells, splitting them in two halves. A cell of level k has size h k = h 0 2 k. k =1 k =2 k =3 k =3 x h 0 M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

7 Adaptivity 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

8 Adaptivity Error indicators for hyperbolic conservation laws geometric: gradient, curvature,... (which component?) Richardson: ok with CLAWPACK grids, does not distinguish a shock from a contact (Ohlberger ) estimators based on Kružkov theory, so only scalar (Karni,Kurganov,Petrova 2002) better, but not very localized (Puppo, M.S.) numerical entropy production: no problem for systems, distinguish contacts and shocks, cheap, info localized in one cell spatial entropy dissipation for entropy stable fluxes (in progress with Mishra, Fiordholm, Puppo): same as above, possibly cheaper M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

9 Adaptivity Entropy inequality u t + f (u) x = 0 conservation law Entropy pair: (regular) functions (η(u), ψ(u)) such that η is convex and η (u)f (u) = ψ (u) or T ψ = T η f M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

10 Entropy inequality Adaptivity u t + f (u) x = 0 conservation law Entropy pair: (regular) functions (η(u), ψ(u)) such that η is convex and η (u)f (u) = ψ (u) or T ψ = T η f In particular for smooth flows η(u) t + ψ(u) x = 0 On a general flow, the entropy inequality selects the unique physically relevant weak solution η(u) t + ψ(u) x 0 M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

11 Adaptivity Semidiscrete finite volume scheme t u + x f (u) = 0 t η(u) + x ψ(u) 0 Update the cell averages with Runge-Kutta scheme u n+1 j = u n j λ where the fluxes are computed as ν i=0 A b ( ) b i F (i) j+1/2 F (i) j 1/2 F (i) j+1/2 = F (u(i), j+1/2, u(i),+ j+1/2 ), i.e. using F consistent with f (u) and non-oscillatory reconstruction procedure to get u (i),± j+1/2 from the cell averages of the stage values u (i) j i 1 = u n j λ k=1 ( ) a ik F (k) j+1/2 F (k) j 1/2 M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

12 Adaptivity Numerical entropy production At each Runge-Kutta stage, t u + x f (u) = 0 t η(u) + x ψ(u) 0 Ψ (i) j+1/2 = Ψ(u(i), j+1/2, u(i),+ j+1/2 ) using the same reconstructions computed for F j+1/2 and a numerical entropy flux consistent with ψ(u). Numerical entropy production in the j th cell { Sj n = 1 ( ) η u n+1 j η ( u n ) ν ( ) } j + λ b i Ψ (i) t j+1/2 Ψ(i) j 1/2 i=1 No extra reconstructions are needed! Note: for order higher than 2, replace η ( u n ) j with a cell average η(un ) j computed via quadrature, using the reconstruction to evaluate u n (x) at the nodes. M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

13 Rate of convergence Adaptivity Consider a semidiscrete scheme of order p as before. Theorem If the numerical entropy flux Ψ is consistent with the exact flux ψ(u), then S n j = O(h p ) for h 0, provided that the solution is smooth in the j th cell. In general, Sj n = O(1/h). Idea of proof. Sj n is the residual of a finite volume scheme for t η(u) + x ψ(u) = 0 Remark On a shock: S n j C h M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

14 Adaptivity Gas dynamics equations: single rarefaction S n j = O(h 2 ) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

15 Adaptivity Gas dynamics equations: single contact S n j = O(1) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

16 Adaptivity Gas dynamics equations: single shock S n j C/h cells S M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

17 Adaptivity Choose your favourite indicator and loop until final time: (next timestep) compute u n+1 j and indicator Sj n j : { Sj n > S ref? h j > h min NO maybe save YES (locally) recompute locally refine coarsen where Sj n < S coa M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

18 Adaptivity Numerical entropy production in the Sod test With the second order scheme (Heun + KT flux + minmod), M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

19 Timestepping 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

20 Timestepping Imposing the CFL globally CFL = k : t λh k = λh 0 2 k = t = λh 0 2 kmax t n+4 t t n+3 t n+2 t n+1 t n represents i b i F (i) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

21 Timestepping Local timestepping Choosing t globally ( t-mode from now on): easier to program wastes CPU resources for big cells, where the solution is well approximated unneeded numerical diffusion if many cells have sub-optimal (local) mesh ratio M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

22 Local timestepping Timestepping Choosing t globally ( t-mode from now on): easier to program wastes CPU resources for big cells, where the solution is well approximated unneeded numerical diffusion if many cells have sub-optimal (local) mesh ratio CFL-mode: local timestepping, multirate Runge-Kutta Each cell advances with different stepsizes t synchro t k = λh k = λh 0 2 k t 1 t 2 t3 x M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

23 Timestepping Review of local timestepping Most of the work on local timestepping was motivated by approximating (on uniform grids) conservation laws with strong spatial variations of the velocity. Osher-Sanders (1983) First order Dawson-Kirby (2001), Kirby (2003) Second order, TVD property Tang-Warnecke(2006) Arbitrary order, but not conservative Muller-Stiriba(2007) and reference therein for multiresolution schemes multiresolution, based on wavelets. Kværno-Rentrop (1999) Multirate Runge-Kutta Costantinescu-Sandu (1999) Partitioned Runge-Kutta M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

24 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

25 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

26 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

27 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

28 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

29 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

30 Timestepping First order scheme on nonuniform grid local timestepping t k = λh k t n+1 t u n+1 1 u n+1 2 u n+1 3 u n+1 4 u n+1 5 u n+1 6 t n u n 1 u n 2 u n 3 u n 4 u n 5 u n 6 x At each intermediate step, we operate on all cells smaller than a given size (computable by fast shift operations). M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

31 Timestepping Let s test the performance of nonuniform grids 1 build a nonuniform grid 2 leave it fixed in time 3 make different kinds of waves travel along them, crossing grid discontinuities. M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

32 M. Semplice et al. (Univ. Torino) u (x) Grid = [cos(2πx)] adaptivity 4 28 th Apr / 46 Timestepping Smooth solution: linear transport of [sin(2πx)] 4

33 Timestepping Contact wave (linear transport) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

34 Timestepping Shock (Burgers equation) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

35 Timestepping Shock (Burgers equation) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

36 Timestepping Errors on uniform/nonuniform grids M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

37 Timestepping Smooth solution (Euler equations) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

38 Timestepping Smooth solution (Euler equations) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

39 Timestepping Rarefaction (Euler equations) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

40 Timestepping Contact wave (Euler equations) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

41 Timestepping Shock wave (Euler equations) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

42 Timestepping Local characteritic projection Zoom of a solution to the Lax shock tube: reconstruction in conservative variables characteristic projection everywhere characteristic projection locally (driven by entropy production) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

43 Sod test Timestepping M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

44 Lax test (1) Timestepping M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

45 Lax test (2) Timestepping M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

46 Local recomputation 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

47 Local recomputation Recomputation after refinement uniform t When integrating in the uniform t mode, it is enough to recompute F (1) at 2 neighbouring cell boundaries and u (1) for one neighbouring cell recompute F (2) at 4 neighbouring cell boundaries and u (1) for two neighbouring cell u (1) u n+1 original recomputed t x

48 Local recomputation Recomputation after refinement uniform t When integrating in the uniform t mode, it is enough to recompute F (1) at 2 neighbouring cell boundaries and u (1) for one neighbouring cell recompute F (2) at 4 neighbouring cell boundaries and u (1) for two neighbouring cell u (1) u n+1 original recomputed t x

49 Local recomputation Recomputation after refinement uniform t When integrating in the uniform t mode, it is enough to recompute F (1) at 2 neighbouring cell boundaries and u (1) for one neighbouring cell recompute F (2) at 4 neighbouring cell boundaries and u (1) for two neighbouring cell u (1) u n+1 original recomputed t x M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

50 Local recomputation Recomputation after refinement local timestepping Following the domain of dependence is hard, computationally expensive t n+1 t u n+1 1 u n+1 2 u n+1 3 u n+1 4 u n+1 5 u n+1 6 t n u n u n 1 u n 2 u n 3 u n 4 u n 5 u n 1,0 u n 1,1 6 x Affects on up to 2 lmax cells per side! And it gets worse for higher order shemes, because of the wider reconstruction stencils and of the stage values M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

51 Local recomputation Recomputation after refinement local timestepping Following the domain of dependence is hard, computationally expensive (and unneeded since we work with an adaptive framework) The error indicator was happy about the solution away from the first cell, so only recompute red quantitities in the diagram: t conservation! t n+1 t n x Conservation requires that the buffer ends with cell averages! M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

52 Reconstruction and fluxes 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

53 Reconstruction and fluxes Shock-acoustic test problem (1) Using minmod limiter M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

54 Reconstruction and fluxes Shock-acoustic test problem (1) 2 2 CT limiter from Cada, Torrilhon J. Comput. Phys. (2009) M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

55 Two space dimensions 1 Locally refined grids 2 Adaptivity 3 Timestepping 4 Local recomputation 5 Reconstruction and fluxes 6 Two space dimensions M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

56 Two space dimensions Atmospheric instability: the problem Scalar conservation law, mimicking the mixing of a cold and hot front: with initial data and velocity u t + (v(x, y)u) = 0 on Ω = [ 4, 4] 2 u 0 (x, y) = tanh(y/2) v(x, y) = [ y r f.385, x ] f r.385 with r = x 2 + y 2 and f = tanh(r)/(cosh(r)) 2. M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

57 Two space dimensions Atmospheric instability: the solution M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46

58 Two space dimensions Euler equations: 2D Riemann problem M. Semplice et al. (Univ. Torino) Grid adaptivity 28 th Apr / 46