Filtered scheme and error estimate for first order Hamilton-Jacobi equations

Transcription

1 and error estimate for first order Hamilton-Jacobi equations Olivier Bokanowski 1 Maurizio Falcone Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) 2 SAPIENZA - Università di Roma Departmento di Matematica 4 Nov, 214, Rome and error estimate for first order Hamilton-Jacobi equations

2 Outline and error estimate for first order Hamilton-Jacobi equations

3 Time dependent HJ equation We are interested in computing the approximation of viscosity solution of Hamilton-Jacobi (HJ) equation: { t v + H(x, v) =, (t, x) (, T ) R d, v(, x) = v (x), x R d. (1) (A1) H(x, p) is continuous in all its variables. (A2) v (x) is Lipschitz continuous. We aim to propose new higher order schemes, and prove their properties of consistency, stability and convergence. and error estimate for first order Hamilton-Jacobi equations

4 Several schemes have been developed: Finite difference schemes (Crandall-Lions(84), Sethian(88), Osher/Shu(91), Tadmor/Lin()). Discontinuous Galerkin approach (Hu/Shu(99), Li/Shu(5), Bokanowski/Chang/Shu(11,13,14), Cockburn(), Capuzzo Dolcetta(83,89,9)). Semi-Lagrangian schemes (Falcone(94,9)/ Ferretti(3,1,13)/Carlini(3,4)). Finite Volume schemes (Kossioris/Makridakis/Souganidis(99), Kurganov/Tadmor()). and error estimate for first order Hamilton-Jacobi equations

5 Monotone scheme Discretization: Let t > denotes the time steps and x > a mesh step, t n = n t, n [,..., N], N N and x j = j x, j Z. For the given function u(x). Finite difference scheme (FD) Crandall-Lions (84) : Let S M be a monotone FD scheme u n+1 (x j ) S M (u n j ) := u n j t h M (x j, D u n j, D + u n j ) (2) with D ± u n j := ± un j±1 u n j x. Assumptions on h M : (A3) h M is Lipschitz continuos function. (A4) (Consistency) x, u, h M (x, v, v) = H(x, v). (A5) (Monotonicity) for any functions u, v, u v = S M (u) S M (v). and error estimate for first order Hamilton-Jacobi equations

6 Monotone scheme Discretization: Let t > denotes the time steps and x > a mesh step, t n = n t, n [,..., N], N N and x j = j x, j Z. For the given function u(x). Finite difference scheme (FD) Crandall-Lions (84) : Let S M be a monotone FD scheme u n+1 (x j ) S M (u n j ) := u n j t h M (x j, D u n j, D + u n j ) (2) with D ± u n j := ± un j±1 u n j x. Assumptions on h M : (A3) h M is Lipschitz continuos function. (A4) (Consistency) x, u, h M (x, v, v) = H(x, v). (A5) (Monotonicity) for any functions u, v, u v = S M (u) S M (v). and error estimate for first order Hamilton-Jacobi equations

7 Consistency error estimate: For any v C 2 ([, T ] R), there exists a constant C M independent of x such that E S M (v)(t, x) := v(t + t, x) SM (v(t,.))(x) ( v t(t, x) + H(x, v x(t, x))) t ( E S M (v)(t, x) C M t ttv + x xxv ). Theorem (Crandall-Lions (84)) Let Hamiltonian H and initial data v be Lipschitz continous. Let the monotone finite difference scheme (2) (with numerical hamiltonian h M satisfies (A3)-(A5)). Let vi n := v(t n, x i ), where v is the exact solution of (1). Then there is a constant C such that for any n T / t and i Z, we have v n (x i ) u n (x i ) C x. (3) for t, x = c t. and error estimate for first order Hamilton-Jacobi equations

8 Consistency error estimate: For any v C 2 ([, T ] R), there exists a constant C M independent of x such that E S M (v)(t, x) := v(t + t, x) SM (v(t,.))(x) ( v t(t, x) + H(x, v x(t, x))) t ( E S M (v)(t, x) C M t ttv + x xxv ). Theorem (Crandall-Lions (84)) Let Hamiltonian H and initial data v be Lipschitz continous. Let the monotone finite difference scheme (2) (with numerical hamiltonian h M satisfies (A3)-(A5)). Let vi n := v(t n, x i ), where v is the exact solution of (1). Then there is a constant C such that for any n T / t and i Z, we have v n (x i ) u n (x i ) C x. (3) for t, x = c t. and error estimate for first order Hamilton-Jacobi equations

9 High order schemes Let S A denote a high order (possibly unstable) scheme: S A (u n )(x) := u n (x) τh A (x, D k, u,..., D u n (x), D + u n (x),..., D k,+ u n (x)) (4) where h A corresponds to a "high order" numerical Hamiltonian D l,± u(x) := ± u(x±l x) u(x) x for l = 1,..., k. Assumptions on S A : (A6) h A is a Lipschitz continuous function. (A7)(High order consistency) There exists k 2, l [1,..., k], for any v = v(t, x)of class C l+1, there exists C A,l, E S A(v)(t, x) := v(t + t, x) SA (v(t,.))(x) ( v t(t, x) + H(x, v x(t, x))) t ( E S A(v)(t, x) C A,l t l l+1 t v + x l x l+1 v ). and error estimate for first order Hamilton-Jacobi equations

10 It is known (Godunov s Theorem) that a monotone scheme can be at most of first order. Therefore it is needed to look for non-monotone schemes. The difficulty is then to combine non-monotony with a convergence to viscosity solution of (1), and also obtain error estimates. This is the core of the present work. In our approach we adapt an idea of Froese and Oberman (13) (for second order HJ equations) to treat mainly the case of evolutive first order PDEs. and error estimate for first order Hamilton-Jacobi equations

11 It is known (Godunov s Theorem) that a monotone scheme can be at most of first order. Therefore it is needed to look for non-monotone schemes. The difficulty is then to combine non-monotony with a convergence to viscosity solution of (1), and also obtain error estimates. This is the core of the present work. In our approach we adapt an idea of Froese and Oberman (13) (for second order HJ equations) to treat mainly the case of evolutive first order PDEs. and error estimate for first order Hamilton-Jacobi equations

12 It is known (Godunov s Theorem) that a monotone scheme can be at most of first order. Therefore it is needed to look for non-monotone schemes. The difficulty is then to combine non-monotony with a convergence to viscosity solution of (1), and also obtain error estimates. This is the core of the present work. In our approach we adapt an idea of Froese and Oberman (13) (for second order HJ equations) to treat mainly the case of evolutive first order PDEs. and error estimate for first order Hamilton-Jacobi equations

13 Filtered function Instead of the Froese and Oberman s filter function i.e. F (x) = sign(x) max(1 x 1, ), we used new filter function i.e. F (x) := x 1 x 1 : Froese and Oberman s filter function and new filter function. and error estimate for first order Hamilton-Jacobi equations

14 Filter scheme: The scheme we propose is then u n+1 j ( S F (ui n ) := S M (uj n ) + ɛ tf ) S A (u n j ) S M (u n j ) ɛ t (5) with a proper initialization of u i. Where ɛ = ɛ( t, x) > is the switching parameter that will satisfy lim ɛ =. ( t, x) More precision on the choice of ɛ will be given later on. and error estimate for first order Hamilton-Jacobi equations

15 Consistency error estimate For any regular function v = v(t, x), for all x R and t [, T ], we have E S F (v)(t, x) = v(t + t, x) S F (v(t,.))(x) ( v t + H(x, v x )) t ( ) (6) C M t tt v + x xx v + ɛc t. Definition (ɛ-monotonicity) is ɛ-monotone i.e. For any functions u, v, u v = S(u) S(v) + Cɛ x, where C is independent of ɛ. and error estimate for first order Hamilton-Jacobi equations

16 Consistency error estimate For any regular function v = v(t, x), for all x R and t [, T ], we have E S F (v)(t, x) = v(t + t, x) S F (v(t,.))(x) ( v t + H(x, v x )) t ( ) (6) C M t tt v + x xx v + ɛc t. Definition (ɛ-monotonicity) is ɛ-monotone i.e. For any functions u, v, u v = S(u) S(v) + Cɛ x, where C is independent of ɛ. and error estimate for first order Hamilton-Jacobi equations

17 Limiter correction: The filter scheme works fine for linear PDEs but may switch to first order for nonlinear PDEs in particular when some viscosity aspect occurs. Therefore we have added a limiter correction when some viscous behavior is detected. 1D front propagation case: H(x, v x ) := max a A (f (x, a)v x ). we say that a viscosity aspect occurs at a minimum x i if min a f (x j, a) and max a f (x j, a). (7) In that case, the limiter is such that the scheme iterate will not go below the local minima around the current point: u n+1 j u min,j := min(u n j 1, u n j, u n j+1), (8) and error estimate for first order Hamilton-Jacobi equations

18 and, in the same way, also demand that u n+1 j u max,j := max(u n j 1, u n j, u n j+1). (9) If we consider the high-order scheme to be of the form (4) then, at points where holds, the limiting process amounts to replace h A j by h A j such that: h A i ( ( := max min h A (u n ) j, un j ) ) u min,j, un j u max,j t t 2D Limitter where u n+1 ij = min(max(s A (u n ) ij, uij min ), uij max ), uij min = min(uij n, ui±1,j, n ui,j±1) n and uij max = max(uij n, ui±1,j, n ui,j±1) n. and error estimate for first order Hamilton-Jacobi equations

19 Convergence Theorem Theorem Assume (A1)-(A2), and v bounded. We assume also that S M satisfies (A3)-(A5), and F 1. Let u n denote the filtered scheme (5). Let vj n := v(t n, x j ) where v is the exact solution of (1). Assume < ɛ c x (1) for some constant c >. (i) The scheme u n satisfies the Crandall-Lions estimate u n v n C x, n =,..., N. (11) for some constant C independent of x. and error estimate for first order Hamilton-Jacobi equations

20 Theorem ((Cont.)) (ii) (First order convergence for classical solutions.) If furthermore the exact solution v belongs to C 2 ([, T ] R), and ɛ c x (instead of (1)). Then it holds u n v n C x, n =,..., N, (12) for some constant C independent of x. and error estimate for first order Hamilton-Jacobi equations

21 Theorem ((Cont.)) (iii) (Local high-order consistency.) Let N be a neighborhood of a point (t, x) (, T ) R. Assume that S A is a high order scheme satisfying (A7) for some k 2. Let 1 l k and v be a C l+1 function on N. Assume that ( ) (C A,1 + C M ) v tt τ + v xx x ɛ. (13) Then, for sufficiently small t n t, x j x, t, x, it holds S F (v n ) j = S A (v n ) j and, in particular, a local high-order consistency error for the filtered scheme S F : E S F (v n ) j E S A(v n ) j = O( x l ) (the consistency error E S A is defined as before). and error estimate for first order Hamilton-Jacobi equations

22 Proof. (i) Let w n+1 j w j = S M (w n ) j be defined with the monotone scheme only, with = v (x j ) = uj. By definitions, u n+1 j w n+1 j = S M (u n ) j S M (w n ) j + ɛ tf (.) Hence, by using the monotonicity of S M, max u n+1 j w n+1 j j and by recursion, for n N, max j u n j w n j + ɛ t, max uj n wj n ɛn t T ɛ. j On the other hand, by Crandall and Lions, an error estimate holds for the monotone scheme: max wj n vj n C x, j and error estimate for first order Hamilton-Jacobi equations

23 (Cont.) for some C. By summing up the previous bounds, we deduce max uj n vj n C x + T ɛ, j and together with the assumption on ɛ, it gives the desired result. (ii) Let E n j := v n+1 j S M (v n ) j. If the solution is C 2 regular with bounded t second order derivatives, then the consistency error is bounded by E n j C M ( t + x). (14) Hence u n+1 j v n+1 j = S M (u n ) j S M (v n ) j + te n j + tɛf (.) u n v n + t E n + tɛ. and error estimate for first order Hamilton-Jacobi equations

24 (Cont.) By recursion, for n t T, u n v n u v + T ( max k N 1 Ek + ɛ). By assumption on ɛ, the bound and the fact that t = O( x) (CFL condition) we get the desired result. (iii) To prove that S F (v n ) j = S A (v n ) j, one has to check that as ( t, x). S A (v n ) j S M (v n ) j ɛ t 1 and error estimate for first order Hamilton-Jacobi equations

25 (Cont.) By using the consistency error definitions, S A (v n ) j S M (v n ) j t = v n+1 j ( v n+1 j Hence the desired result follows. S A (v n ) j t + v t (t n, x j ) + H(x j, v x (t n, x j )) S M (v n ) j )+v t (t, x)+h(x j, v x (t n, x j ))) t E S A(v n ) j + E S M (v n ) j (C A,1 + C M )( t v tt + x v xx ) and error estimate for first order Hamilton-Jacobi equations

26 Filtered Semi-Lagrangian scheme Let us consider H(x, p) := min max b B a A { f (x, a, b).p l(x, a, b)}, where A R m and B R n are non-empty compact sets (with m, n 1), f : R d A B R d and l : R d A B R are Lipschitz continuous w.r.t. x: L, (a, b) A B, x, y: max( f (x, a, b) f (y, a, b), l(x, a, b) l(y, a, b) ) L x y. and error estimate for first order Hamilton-Jacobi equations

27 Let [u] denote the P 1 -interpolation of u in dimension one on the mesh (x j ), i.e. x [x j, x j+1 ] [u](x) := x j+1 x x u j + x x j x u j+1. Then a monotone SL scheme can be defined as follows: ( S M (u n ) j := min max [u n ] ( x j + tf (x j, a, b) ) ) + tl(x j, a, b). (15) a A b B A filtered scheme based on SL can then be defined by (5) and (15). Convergence result as well as error estimates could also be obtained in this framework. (For error estimates for the monotone SL scheme, we refer to FF 14 and Soravia 98) and error estimate for first order Hamilton-Jacobi equations

28 Tuning of the parameter ɛ In order to be in the convergence setting ɛ should be small enough i.e. ɛ O( x). Thus we have, Upper bound ɛ C x, with constant C >. The scheme ) u n+1 i ( S = S M (ui n A (ui n ) S M (ui n ) ) + ɛ tf ɛ t has to switch to high order scheme when some regularity is detected, i.e., when (S A (ui n ) S M (ui n )) ɛ t = (h A (ui n ) h M (ui n )) ɛ 1. If we assume that v is regular enough, and error estimate for first order Hamilton-Jacobi equations

29 Tuning of the parameter ɛ In order to be in the convergence setting ɛ should be small enough i.e. ɛ O( x). Thus we have, Upper bound ɛ C x, with constant C >. The scheme ) u n+1 i ( S = S M (ui n A (ui n ) S M (ui n ) ) + ɛ tf ɛ t has to switch to high order scheme when some regularity is detected, i.e., when (S A (ui n ) S M (ui n )) ɛ t = (h A (ui n ) h M (ui n )) ɛ 1. If we assume that v is regular enough, and error estimate for first order Hamilton-Jacobi equations

30 Tuning of the parameter ɛ In order to be in the convergence setting ɛ should be small enough i.e. ɛ O( x). Thus we have, Upper bound ɛ C x, with constant C >. The scheme ) u n+1 i ( S = S M (ui n A (ui n ) S M (ui n ) ) + ɛ tf ɛ t has to switch to high order scheme when some regularity is detected, i.e., when (S A (ui n ) S M (ui n )) ɛ t = (h A (ui n ) h M (ui n )) ɛ 1. If we assume that v is regular enough, and error estimate for first order Hamilton-Jacobi equations

31 Tuning of the parameter ɛ In order to be in the convergence setting ɛ should be small enough i.e. ɛ O( x). Thus we have, Upper bound ɛ C x, with constant C >. The scheme ) u n+1 i ( S = S M (ui n A (ui n ) S M (ui n ) ) + ɛ tf ɛ t has to switch to high order scheme when some regularity is detected, i.e., when (S A (ui n ) S M (ui n )) ɛ t = (h A (ui n ) h M (ui n )) ɛ 1. If we assume that v is regular enough, and error estimate for first order Hamilton-Jacobi equations

32 by using Taylor series expansions and the high order property we have h A (v) i = H(v x (x i )) + O( x 2 ), and also, Dv ± i = v x (x i ) ± 1 2 v xx(x i ) x + O( x 2 ). h M (x i, Dv i, Dv + i ) = H(v x (x i ))+ 1 ( h M ) 2 v (v) i (v) i xx(x i ) x Dv + hm Dv +O( x 2 ). Therefore, 1 2 v xx(x i ) h M (v) i Dv + hm (v) i Dv x ɛ. and error estimate for first order Hamilton-Jacobi equations

33 by using Taylor series expansions and the high order property we have h A (v) i = H(v x (x i )) + O( x 2 ), and also, Dv ± i = v x (x i ) ± 1 2 v xx(x i ) x + O( x 2 ). h M (x i, Dv i, Dv + i ) = H(v x (x i ))+ 1 ( h M ) 2 v (v) i (v) i xx(x i ) x Dv + hm Dv +O( x 2 ). Therefore, 1 2 v xx(x i ) h M (v) i Dv + hm (v) i Dv x ɛ. and error estimate for first order Hamilton-Jacobi equations

34 Tuning of the parameter ɛ Bounds for ɛ : Upper bound: ɛ C x, with constant C > ( to have error estimates and convergence ). Lower bound: 1 2 v xx(x i ) x ɛ. ( to have high-order behavior ) hm (v) i Dv + hm (v) i Dv Typically the choice ɛ := C x is made, for some constant C of the order of v xx L. and error estimate for first order Hamilton-Jacobi equations

35 Tuning of the parameter ɛ Bounds for ɛ : Upper bound: ɛ C x, with constant C > ( to have error estimates and convergence ). Lower bound: 1 2 v xx(x i ) x ɛ. ( to have high-order behavior ) hm (v) i Dv + hm (v) i Dv Typically the choice ɛ := C x is made, for some constant C of the order of v xx L. and error estimate for first order Hamilton-Jacobi equations

36 Tuning of the parameter ɛ Bounds for ɛ : Upper bound: ɛ C x, with constant C > ( to have error estimates and convergence ). Lower bound: 1 2 v xx(x i ) x ɛ. ( to have high-order behavior ) hm (v) i Dv + hm (v) i Dv Typically the choice ɛ := C x is made, for some constant C of the order of v xx L. and error estimate for first order Hamilton-Jacobi equations

37 Eikonal equation in 1D Example 1. { vt + v x =, t >, x ( 2, 2), v(, x) = v (x), := max(, 1 x 2 ) 4, x ( 2, 2), with periodic boundary condition on ( 2, 2), terminal time T =.3 Filter ɛ = 5 x CFD ENO2 M N L 2 error order L 2 error order L 2 error order E E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (2nd order) scheme with RK2 in time. and error estimate for first order Hamilton-Jacobi equations

38 1.5 t= 1.5 t= t=.3 Exact Exact Exact Scheme Scheme Scheme Initial data (left), and plots at time T =.3, by Central finite difference scheme - middle - and - right (M = 16 mesh points). and error estimate for first order Hamilton-Jacobi equations

39 Eikonal equation in 1D Example 2. { vt + v x =, t >, x ( 2, 2), v(, x) = v (x), := max(, 1 x 2 ) 4, x ( 2, 2), with periodic boundary condition on ( 2, 2), terminal time T =.3 Errors Filter ɛ = 5 x CFD ENO2 M N L 2 error order L 2 error order L 2 error order E E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (2nd order) scheme with RK2 in time. and error estimate for first order Hamilton-Jacobi equations

40 1.5 t= 1.5 t= t=.3 Exact Exact Exact Scheme Scheme Scheme Initial data (left), and plots at time T =.3, by Central finite difference scheme - middle - and - right (M = 16 mesh points). and error estimate for first order Hamilton-Jacobi equations

41 1D steady equation Example 3. (as in Abgrall [6]) x := { vx = f (x), on (, 1) v() = v(1) = f (x) := 3x 2 + a, a := 1 2x3 2x 1, Filter (ɛ = 5 x) CFD ENO M L error order L error order L error order E inf E E inf E E inf E E inf E-3 1. Table : L Errors for Filter scheme, CFD scheme, and RK2-2nd order ENO scheme. Filter can stabilize an otherwise unstable scheme and error estimate for first order Hamilton-Jacobi equations

42 Advection with obstacle Example 4. (as in Bokanowski et al. [9]) { min(vt + v x, g(x)) =, t >, x [ 1, 1], u(, x) =.5 + sin(πx) x [ 1, 1], with periodic boundary condition, g(x) = sin(πx), and where the terminal time T =.5. Errors Filter ɛ = 5 x CFD ENO2 M N L error order L error order L error order E E E E E E E E E E E E E E E Table : Local L errors for filter scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time. and error estimate for first order Hamilton-Jacobi equations

43 1.5 t= 1.5 t= t=.5 Exact Exact Exact Scheme Scheme Scheme Initial data (left), and plots at time T =.3 - middle and T =.5- right by filtered scheme. and error estimate for first order Hamilton-Jacobi equations

44 Advection in 2D Example 5. We consider the domain Ω = ( 2.5, 2.5) 2 and { vt yv x + xv y =, (x, y) Ω, t >, v(, x, y) = v (x, y) = v(, x, y) = v (x, y) :=.5.5 max(, 1 (x 1)2 y 2 ) 4 1 r 2 r =.5 and with Dirichlet boundary condition v(t, x) =.5, x Ω, T := π 2. Filter (ɛ = 2 x) CFD ENO2 Mx = Nx N L 2 error order L 2 error order L 2 error order E-1-5.5E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time. and error estimate for first order Hamilton-Jacobi equations

45 Initial data (left), and plots at time T = π/2, by (M = 16 mesh points). and error estimate for first order Hamilton-Jacobi equations

46 Eikonal in 2D Example 6. In this example we consider HJB equation with smooth initial data and Ω = ( 3, 3) 2 { vt + v =, (x, y) Ω, t >, v(, x, y) = v (x, y) = v (x, y) :=.5.5 max(, 1 (x 1)2 y 2 ) 4, where. is the Euclidean norm and r =.5 with Drichlet boundary conditions. 1 r 2 Filter (ɛ = 4 x) CFD ENO2 Mx = Nx N L 2 error order L 2 error order L 2 error order E E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time. and error estimate for first order Hamilton-Jacobi equations

47 y y t= t= Numerical front exact front y x x 1 t=.6 3 t=.6 Numerical front exact front y x x and error estimate for first order Hamilton-Jacobi equations

48 Eikonal in 2D Example 7. Ω = ( 3, 3) 2 { ( v(, x, y) = v (x, y) =.5.5 max max(, 1 (x 1)2 y 2 1 r 2 ) 4, max(, 1 (x+1)2 y 2 ) ). 4 1 r 2 where. is the Euclidean norm and A ± := (±1, ) with Drichlet boundary conditions and CFL condition µ =.37 Filter (ɛ = 2 x) CFD ENO2 Mx = Nx N L 2 error order L 2 error order L 2 error order E E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time. and error estimate for first order Hamilton-Jacobi equations

49 y 1.5 t= 3 2 t= Numerical front exact front y x x 1 t= t=.6 Numerical front exact front.5 1 y y x and error estimate x for first order Hamilton-Jacobi equations

50 Example 8: Ω = ( 3, 3) 2 v t + yv x + xv y v =, (x, y) Ω, ( t >, v(, x, y) = v (x, y) =.5.5 max max(, 1 (x 1)2 y 2 1 r 2 ) 4, max(, 1 (x+1)2 y 2 ) ). 4 1 r 2 Filter (ɛ = 1 x) CFD ENO2 Mx = Nx N L 2 error order L 2 error order L 2 error order E E E E E E E E E E E E E E E Table : L 2 errors for filter scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time. and error estimate for first order Hamilton-Jacobi equations

51 y y t= t= Numerical front exact front y x x 1.8 t=.6 3 t=.6 Numerical front exact front y x x and error estimate for first order Hamilton-Jacobi equations

52 A general and simple presentation of filtered scheme and easy to implement. Convergence of filtered scheme is confirmed. Error estimate is of O( x) and numerically observed that O( x 2 ) behavior in smooth regions. Remark : We propose a general strategy of taking a good scheme (like ENO second order) but for which there is no convergence proof, and use the filter to assure convergence and error estimate (the theoretical x as for the monotone scheme), and numerically show that we almost keep the same precision as ENO (i.e. second order) on basic linear and non linear examples. and error estimate for first order Hamilton-Jacobi equations

53 A general and simple presentation of filtered scheme and easy to implement. Convergence of filtered scheme is confirmed. Error estimate is of O( x) and numerically observed that O( x 2 ) behavior in smooth regions. Remark : We propose a general strategy of taking a good scheme (like ENO second order) but for which there is no convergence proof, and use the filter to assure convergence and error estimate (the theoretical x as for the monotone scheme), and numerically show that we almost keep the same precision as ENO (i.e. second order) on basic linear and non linear examples. and error estimate for first order Hamilton-Jacobi equations

54 A general and simple presentation of filtered scheme and easy to implement. Convergence of filtered scheme is confirmed. Error estimate is of O( x) and numerically observed that O( x 2 ) behavior in smooth regions. Remark : We propose a general strategy of taking a good scheme (like ENO second order) but for which there is no convergence proof, and use the filter to assure convergence and error estimate (the theoretical x as for the monotone scheme), and numerically show that we almost keep the same precision as ENO (i.e. second order) on basic linear and non linear examples. and error estimate for first order Hamilton-Jacobi equations

55 [1] O. Bokanowski, M. Falcone, and S. Sahu,"s for some first order HJ equations", work in progress. [2] M. G. Crandall and P.L. Lions, Two approximations of solution of Hamilton-Jacobi equations. Comput. Method Appl. Mech. Engrg., 195: ,1984. [3] G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal., 4: , [4] B. D. Froese and A. M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation. SIAM J. Numer. Anal., 51: ,213. [5] M. Falcone, and R. Ferretti, Semi-Lagrangian Approximation schemes for linear and Hamilton-Jacobi equations, SIAM-Society for Industrial and Applied Mathematics, Philadelphia, 214. and error estimate for first order Hamilton-Jacobi equations

56 [6] R. Abgrall. Construction of simple stable and convergent high order schemes for steady first order HJ equations. SIAM J. Comput.,31(4): , 29. [7] S. Osher, C.W. Shu,High order essential non-oscillatory schemes for Hamilton-Jacobi equation. SIAM J. Numer. Anal., 4:97-922, [8] O. Bokanowski, M. Falcone, R. Ferretti, L. Grüne, D. Kalise, H. Zidani, Value iteration convergence of epsilon-monotone schemes for stationary HJ equations. Preprint HAL 214. [9] O. Bokanowski, Y. Cheng, and C-W Shu, A discontinuous Galerkin scheme for front propagation with obstacle. Numer. Math.(214) 126:1-31. and error estimate for first order Hamilton-Jacobi equations

57 [1] O. Bokanowski, M. Falcone, and S. Sahu,"s for some first order HJ equations", work in progress. [2] M. G. Crandall and P.L. Lions, Two approximations of solution of Hamilton-Jacobi equations. Comput. Method Appl. Mech. Engrg., 195: ,1984. [3] G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal., 4: , [4] B. D. Froese and A. M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation. SIAM J. Numer. Anal., 51: ,213. [5] M. Falcone, and R. Ferretti, Semi-Lagrangian Approximation schemes for linear and Hamilton-Jacobi equations, SIAM-Society for Industrial and Applied Mathematics, Philadelphia, 214. and error estimate for first order Hamilton-Jacobi equations

58 [6] R. Abgrall. Construction of simple stable and convergent high order schemes for steady first order HJ equations. SIAM J. Comput.,31(4): , 29. [7] S. Osher, C.W. Shu,High order essential non-oscillatory schemes for Hamilton-Jacobi equation.siam J. Numer. Anal., 4:97-922, [8] O. Bokanowski, M. Falcone, R. Ferretti, L. Grüne, D. Kalise, H. Zidani, Value iteration convergence of epsilon-monotone schemes for stationary HJ equations. Preprint HAL 214. [9] O. Bokanowski, Y. Cheng, and C-W Shu, A discontinuous Galerkin scheme for front propagation with obstacle. Numer. Math.(214) 126:1-31. and error estimate for first order Hamilton-Jacobi equations