Approximation des systemes hyperboliques par elements finis continus non uniformes en dimension quelconque

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1 Approximation des systemes hyperboliques par elements finis continus non uniformes en dimension quelconque Jean-Luc Guermond and Bojan Popov Department of Mathematics Texas A&M University Séminaire du Laboratoire Jacques-Louis Lions UPMC 04 Nov 2016

2 Acknowledgments Collaborators: Murtazo Nazarov (Uppsala University) Vladimir Tomov (LLNL) Young Yang (Penn State University) Laura Saavedra (Universidad Politécnica de Madrid) Support:

3 Hyperbolic systems 1 Hyperbolic systems 2 FE approximation 3 Hyperbolic systems + ALE 4 Maximum wave speed Hyperbolic systems

4 Hyperbolic systems The PDEs Hyperbolic tu + r f(u) =0, (x, t) 2 D R +. u(x, 0) = u 0 (x), x 2 D. D open polyhedral domain in R d. f 2C 1 (R m ; R m d ), the flux. u 0,admissibleinitialdata. Periodic BCs or u 0 has compact support (to simplify BCs)

5 Hyperbolic systems Examples Linear transport: f(u) = Burgers equation: f (u) = 1 2 u2 Tra c flow equation: f (u) =v max(1 Bucley-Leverett: f (u) = Shallow water: p-system Euler equations... u = 2 R q 1+d, f(u) = u u u max )u u 2 u 2 +a(1 u) 2, a is constant 1! q T 1 q q R 2 g 2 I (1+d) d d

6 Formulation of the problem Assumptions 9 admissible set A s.t. for all (u l, u r ) 2Athe 1D Riemann problem ( u l if x < tv x (n f(v)) = 0, v(x, 0) = u r if x > 0. has a unique entropy solution u(u l, u r )(x, t) foralln 2 R d, knk`2 =1. There exists an invariant set A A,i.e., A is convex. u(u l, u r )(x, t) 2 A, 8t 0, 8x 2 R, 8u l, u r 2 A.

7 Formulation of the problem Examples of invariant sets Invariant domains are convex for genuinely nonlinear systems (Ho (1979, 1985), Chueh, Conley, Smoller (1973)). Scalar conservation in R d : A =[a, b], 8a apple b 2 R. Euler: A = { >0, e > 0, s a}, 8a 2 R,wheres is the specific entropy. p-system (1D): etc. U =(v, u) T A := {U 2 R + R a apple W 2 (U) apple W 1 (U) apple b}, 8a apple b 2 R where W 1 and W 2 are the Riemann invariants Z 1 p W 1 (U) =u + p 0 (s) ds, and W 2 (U) =u v Z 1 p v p 0 (s) ds.

8 FE approximation 1 Hyperbolic systems 2 FE approximation 3 Hyperbolic systems + ALE 4 Maximum wave speed Hyperbolic systems

9 Approximation (time and space) FE space/shape functions {T h } h>0 shape regular conforming mesh sequence {' 1,...,' I },positive+partitionofunity( P j2{1:i } ' j =1) Ex: P 1, Q 1,Bernsteinpolynomials(anydegree) m i := R D ' i dx, lumpedmassmatrix(m i = P j2i(s i ) R D ' i ' j dx)

10 Approximation (time and space) Algorithm: Galerkin Set u h (x, t) = P I j=1 U j (t)' j (x). Galerkin + lumped mass matrix m Z + r (f(u h ))' i dx =0 D Algorithm: Galerkin + First-order viscosity + Explicit Euler by Un+1 i t Approximate f(u h )by P j2i(s i ) (f(un j ))' j U n+1 i m i t U n i Z + D U n i 0 1 X (f(u n j ))' j A ' i dx + X dij n (Un i U n j ) =0. j2i(s i ) j2i(s i ) How should we choose artificial viscosity d n ij?

11 Approximation (time and space) Algorithm: Galerkin + First-order viscosity + Explicit Euler Introduce Then U n+1 i m i Z c ij = ' i (x)r' j (x) dx. D t U n i = X j c ij f(u j )+d n ij U j Observe that conservation implies P j c ij =0,(partitionofunity) We define d n ii such that P j d n ij =0,(conservation).. Remark Rest of the talk applies to any method that can be formalized as above. (FV, DG, FD, etc.)

12 Approximation (time and space) Algorithm: Galerkin + First-order viscosity + Explicit Euler Observe that conservation implies P j c ij =0and P j d n ij =0. U n+1 i m i t U n i = X j c ij (f(u i ) f(u j )) + dij n (U i + U j ). Try to construct convex combination... U n+1 i = U n i (1 + 2 t m i D ii )+ X j6=i = U n i (1 X j6=i 2 t m i d n ij )+X j6=i t c ij (f(u i ) f(u j )) + dij n m (U i + U j ) i 2 t dij n m i Introduce intermediate states U(U i, U j ) 1 2 (U i + U j )+ c! ij 2dij n (f(u i ) f(u j )) U(U i, U j ):= 1 2 (U i + U j )+ c ij 2dij n (f(u i ) f(u j )).

13 Approximation (time and space) Algorithm: Galerkin + First-order viscosity + Explicit Euler Now construct convex combination U n+1 i = U n i (1 X 2 t dij n m )+X j6=i i j6=i Are the states U(U i, U j )goodobjects? 2 t m i d n ij U(U i, U j )

14 Approximation (time and space) Algorithm: Galerkin + First-order viscosity + Explicit Euler Define n ij = c ij /kc ij k`2 2 R d,(unitvector). f ij (U) :=n ij f(u) isanhyperbolicfluxbydefinitionofhyperbolicity! Then U(U i, U j ):= 1 2 (U i + U j )+ kc ij k`2 2dij n (f ij (U i ) f ij (U j )).

15 Approximation (time and space) Lemma (GP (2015)) Consider the fake 1D Riemann tv x (n ij f(v)) = 0, v(x, 0) = ( U i if x < 0 U j if x > 0. Let max(f, n ij, U i, U j ) be maximum wave speed in 1D Riemann problem Then U(U i, U j )= R v(x, t) dx with fake time t = kc ij k`2 2d ij n,provided kc ij k`2 2d n ij max(f, n ij, U i, U j )= t max(f, n ij, U i, U j ) apple 1 2 Define viscosity coe cient d n ij := max(f, n ij, U i, U j )kc ij k`2, j 6= i.

16 Approximation (time and space) Theorem (GP (2015)) Provided CFL condition, (1 2 t m i D ii ) 0. Local invariance: U n+1 i 2 Conv{U(U n i, Un j ) j 2I(S i )}. Global invariance: The scheme preserves all the convex invariant sets. (Let A be a convex invariant set, assume U 0 2 A, thenu n+1 i 2 A for all n 0.) Discrete entropy inequality for all the entropy pairs (, q): Z m i t ( (Un+1 i ) (U n i )) + D r ( h q(u n h ))' i dx + X i6=j2i(s i ) d ij (U n j ) apple 0.

17 Approximation (time and space) Is it new? Loose extension of non-staggered Lax-Friedrichs to FE. Similar results proved by Ho (1979, 1985), Perthame-Shu (1996), Frid (2001) in FV context and compressible Euler. Some relation with flux vector splitting theory of Bouchut-Frid (2006). Not aware of similar results for arbitrary hyperbolic systems and continuous FE.

18 A priori error estimate for scalar equations: Definition of mollifiers Let >0and = kfk Lip Consider mollifiers! and! 8 1 >< t apple, 3 2 t! (t) := 3 >: 2 apple t apple2, 0 otherwise,! (x) := d l=1! (x l ), x := (x 1,...,x d ). Following Kruskov (1970), define (x, y, t, s) :=! (x y)! (t s), 8(y, s) 2 D [0, T ]. Following Cockburn-Gremaud (1996,1998), define Z t (t) :=! (s) ds. 0

19 A priori error estimate for scalar equations: A useful lemma Lemma (Guermond, Popov ( )) Assume u 0 2 BV ( ). Leteu h : D [0, T ]! R be any approximate solution. Assume that there is aboundedfunctionalonlipschitzfunctionssothat8k 2 [u min, u max], 8 2 Wc 1,1 (D [0, T ]; R + ): Z T Z eu h t + sgn(eu h k)(f(eu h ) f(k)) r dx dt 0 D + k h (eu h (T ) k) h (, T h ) k`1 k h (eu h (0) k) h (, h h) k`1 apple ( ), h where k k`1 h is the discrete L 1 -norm and T T h apple t, 0 h apple t, >0 is a uniform constant. Then the following estimate holds ku(, T ) ũ h (, T )k L 1 ( ) apple c (( + h) u 0 BV ( ) + ) R t0 R where D ( ) dy ds := sup 0appletappleT. (t) Generalization of results by Cockburn-Gremaud (1996) and Bouchut-Perthame (1998) based on Kruskov (1970), Kuznecov (1976).

20 A priori error estimate for scalar equations English translation Control on all the Kruskov entropies ) Convergence estimate. Theorem (Guermond, Popov ( )) Assume u 0 2 BV and f Lipschitz. Let u h be the first-order viscosity solution. Then there is c 0,uniform,suchthatthefollowingholdsifCFLapple c 0 : (i) ku(t ) u h (T )k L 1 ((0,T );L 1 ) apple ch 1 2 if a priori BV estimate on u h. (ii) ku(t ) u h (T )k L 1 ((0,T );L 1 ) apple ch 1 4 otherwise. BV estimate is trivial in 1D (Harten s lemma). BV estimate can be proved in nd on special meshes. Similar results for FV Chainais-Hillairet (1999), Eymard et al (1998) First error estimates for explicit continuous FE method (as far as we know).

21 High-order extension Higher-order in time Use SSP method to get higher-order in time. Strong Stability Preserving methods (SSP), Kraaijevanger (1991) (amazing paper), Gottlieb-Shu-Tadmor (2001), Spiteri-Ruuth (2002) Ferracina-Spijker (2005), Higueras (2005), etc.:

22 High-order extension Higher-order in time The midpoint rule is not SSP Heun s method is SSP w (1) = u n + tl(t n, u n ) w (2) = w (1) + tl(t n + t, w (1) ) u n+1 = 1 2 un w (2). SSPRK(3, 3): w (1) = u n + tl(t n, u n ), z (1) = w (1) + tl(t n + t, w (1) ), w (2) = 3 4 un z(1), z (2) = w (2) + tl(t n t, w (2) ), u n+1 = 1 3 un z(2).

23 High-order extension Remark on SSP SSP is not about positivity, it is about convexity. Let U! S t (U) besspschemebasedoneulerstepu! E t (U), t apple t 0, Let A be a convex set then If Euler step E t (U) isinvariantdomainpreservingina then SSP step S t (U) isinvariantdomainpreservingina SSP methods preserve convex domains that are invariant for forward Euler time stepping (It s all about convexity).

24 High-order extension Higher-order in space: Entropy viscosity Use entropy viscosity (or something else) FCT or other limitation (work in progress)

25 Strong explosion; ent. vis. sol. 1.5 million P 2 nodes (author: Murtazo Nazarov; 1.5 million P 2 nodes)

26 Mach 10 ramp, ent. vis. sol. 1.2 million P 2 nodes (author: Murtazo Nazarov; 1.2 millions P 2 nodes)

27 Mach 3, Step, ent. vis. sol P 1 nodes and viscous sol P 1 nodes

28 Hyperbolic systems + ALE 1 Hyperbolic systems 2 FE approximation 3 Hyperbolic systems + ALE 4 Maximum wave speed Hyperbolic systems

29 ALE formulation Instead of tracking the characteristics (there are too many), we want to move the mesh. ALE formulation Let : R d R +! R d be a uniformly Lipschitz mapping (R d 3 7! (, t) 2 R d invertible on [0, t ]) Let v A (x, t) =@ t ( 1 t (x), t) Arbitrary Lagrangian Eulerian velocity We are going to use v A to move the mesh. Lemma The following holds in the distribution sense (in time) over [0, t ] for every function 2 C0 0(Rd 1 ; R) (with the notation '(x, t) := ( t (x))): t u(x, t)'(x, t) dx = r (u v A f(u))'(x, t) dx. ZR d R d

30 Finite elements Geometric Finite elements Let (T 0 h ) h>0 be a shape-regular sequence of matching meshes. Reference Lagrange finite element ( b K, b P geo, b geo )forgeometry Lagrange nodes {ba i } i2{1:n geo sh } and Lagrange shape functions {b geo i } geo i2{1:n sh } {a n i } i2{1:i geo } collection of all the Lagrange nodes in the mesh Th n j geo : Th n {1:ngeo sh }!{1:I geo } geometric connectivity array Geometric transformation T n K : b K T n K (bx) =! K defined by X i2{1:n geo sh n a } b j geo (i,k) geo i (bx). ) Mesh motion controlled by motion of Lagrange nodes

31 Finite elements Approximating Finite elements Reference finite element ( b K, b P, b )} Shape functions b i (x) 0, P i2{1:n sh } b i (bx) =1 Finite element spaces P(T n h ):={v 2C0 (D n ; R); v K P d (Th n ):=[P(T h n )]d, P m(th n ):=[P(T h n )]m. { n i } i2{1:i } global shape functions in P(T n h ). T n K 2 b P, 8K 2T n h },

32 Finite elements The algorithm Initialization: m 0 i := R R d n i (x) dx u h0 := P i2{1:i } U0 i 0 i 2 P m(t 0 h ) ALE velocity field given: w n = P i2{1:i } Wn i n i Mesh motion: a n+1 i = a n i + tw n (a n i ). Mass update: (do not use m n+1 i = R n+1 D i dx!) Z m n+1 i = m n i + t Update approximation field u n+1 h S n i 2 P d (T n h ), n i (x)r wn (x) dx. m n+1 i U n+1 i t m n i Un i X j2i(s i n ) Z + r R d d n ij Un j X j2{1:i } (f(u n j ) Un j W n j ) n j (x) n i (x) dx =0,

33 Finite elements Definition of d n ij Consider flux g n j (v) :=f(v) v Wn j, j 2{1:I } Consider one-dimensional Riemann tv x (g n j (v) nn ij )=0, (x, t) 2 R R+, v(x, 0) = ( U n i if x < 0 U n j if x > 0. Define d n ij by dij n =max( max(gj n, nn ij, Un i, Un j )kcn ij k`2, max(gn i, nn ji, Un j, Un i )kcn ji k`2 ). Note that max(g n j, nn ij, Un i, Un j )=max( L(f, n n ij, Un i, Un j ) Wn j nn ij, R(f, n n ij, Un i, Un j ) Wn j nn ij ).

34 Conservation and invariant domain property Theorem (GPSY (2015)) The total mass P i2{1:i } mn i Un i is conserved. Provided CFL condition, (1 2 t m n D ii ) 0. i Local invariance: U n+1 i 2 Conv{U(U n i, Un j ) j 2I(S i )}. Global invariance. Let A be a convex invariant set, assume U 0 2 A, then U n+1 i 2 A for all n 0. The scheme preserves all the convex invariant sets. Discrete entropy inequality for any entropy pair (, q) 1t mn+1 i (U n+1 i ) m n i (Un i ) apple X Z r R d j2i(s i n ) X j2i(s i n ) (q(u n j ) d n ij (Un j ) (Un j )Wn j ) n j (x) n i (x) dx Corollary (GPSY (2015)) The scheme preserves constant states (Discrete Global Conservation Law (DGCL))

35 Hyperbolic systems FE approximation Hyperbolic systems + ALE Maximum wave speed 2D u + r ( 12 u 2 ) = 0, u0 (x) = 1S, with := (1, 1)T, S := (0, 1)2 Figure: Burgers equation, mesh. Left: Q1 FEM with 25 contours; Center left: Final Q1 mesh; Center right: P1 FEM with 25 contours; Right: Final P1 mesh.

36 Hyperbolic systems FE approximation Hyperbolic systems + ALE Maximum wave speed Nonconvex flux (KPP u + r f(u) = 0, u0 (x) = kxk `2 < , with f(u) = (sin u, cos u)t Figure: KPP problem, mesh. Left: Q1 FEM with 25 contours; Center left: Final Q1 mesh; Center right: P1 FEM with 25 contours; Right: Final P1 mesh.

37 Euler Compressible Euler, 2D Noh problem, = 5 3 Initial data x 0 (x) =1.0, u 0 (x) = 1 x6=0, p 0 (x) = kxk`2 Q 1 P 1 #dofs L 2 -norm L 1 -norm L 2 -norm L 1 -norm E E E E E E E E E E E Table: Noh problem, convergence test, T = 0.6, CFL = 0.2

38 Compressible Euler, 2D Noh problem, = 5 3 Figure: Noh problem at t =0.6, mesh. From left to right: density field with Q 1 approximation (25 contour lines); mesh with Q 1 approximation; density field with P 1 approximation (25 contour lines); mesh with P 1 approximation.

39 Compressible Euler, 3D Noh problem, = 5 3 Figure: Density cuts for the 3D Noh problem at t =0.6. Figure: 3D Noh problem at t = MPI tasks division.

40 Maximum wave speed 1 Hyperbolic systems 2 FE approximation 3 Hyperbolic systems + ALE 4 Maximum wave speed Hyperbolic systems

41 How to compute local viscosity? d n ij := 2 max(f, n ij, U i, U j )kc ij k`2,forj 6= i. max(f, n ij, U i, U j )ismaxwavespeedforriemannproblem

42 Riemann fan for Euler, p =( 1) e Structure of the Riemann problem (Lax (1957), Bressan (2000), Toro (2009)). Waves 1 and 3 are genuinely nonlinear (either shock or rarefaction) Wave 2 is linearly degenerate (contact) w L =( L, u L, p L ), wl =( L, u, p ), wr =( R, u, p ), w R =( R, u R, p R ),

43 Maximum wave speed bound Euler system, p =( 1) e Given the states U L and U R,wehave 1 = u L a L 1+ (p 1 p L ) < 3 = u R +a R 1+ (p 1 p R ) p L 2 p R 2 where p is the pressure of the intermediate state. Then and define max(u L, U R )=max( 1, 3 ). In practice we just need a good upper bound of p : p p.then max(u L, U R )=max( 1(p ), 3(p ) ).

44 Maximum wave speed bound To avoid computing p,itisacommonpracticetoestimate max( u L + a L, u R + a R ) This estimate is inaccurate and can be wrong. max by

45 Maximum wave speed bound Counter-example 1: 1-wave and the 3-wave are both shocks Toro 2009, L R u L u R p L p R max but max( u L + a L, u R + a R ) 29.97, large overestimation

46 Maximum wave speed bound Counter-example 2: 1-wave is a shock and the 3-wave is an expansion L R u L u R p L p R max but max( u L + a L, u R + a R ) 1.183, large underestimation

47 Definition of p Let p be the zero of R, then 0 1 p B a 1 L + a R 2 (u R u L ) C 1 1 A 2 2 a L pl + a R pr 2 1 Lemma (GP (2016)) We have p < p in the physical range of, 1 < apple 5 3. p is an upper bound on p. min(p L, p R ) apple p apple p (starting guess for cubic Newton alg., GP (2016))

48 Conclusions Continuous finite elements Continuous FE are viable tools to solve hyperbolic systems. Continuous FE are viable alternatives to DG and FV. Continuous FE are easy to implement and parallelize. Exa-scale computing will need simple, robust, methods. Current and future work Convergence analysis, error estimates beyond first-order. Extension to DG. Extension of BBZ to higher-order polynomials (order 3 and higher). Extension of BBZ to systems (Shallow water, Euler). Extension to equations with source terms (Radiative transport, Radiative hydrodynamics).