An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equat

Transcription

1 An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations joint work with Olivier Bokanowski 2 and Maurizio Falcone SAPIENZA - Università di Roma 2 Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) Conference on the occasion of Maurizio Falcone s 6th birthday Rome 4-5, Dec. 24

2 Outline 2 3 4

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. () (A) 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.

4 Several schemes have been developed: Finite difference schemes (Crandall-Lions(84), Sethian(88), Osher/Shu(9), Tadmor/Lin()). Semi-Lagrangian schemes (Capuzzo Dolcetta(83,89,9)), (Falcone(94,9)/ Ferretti-Carlini(3,4,3)). Discontinuous Galerkin approach (Hu/Shu(99), Li/Shu(5), Bokanowski/Cheng/Shu(,3,4), Cockburn(). Finite Volume schemes (Kossioris/Makridakis/Souganidis(99), Kurganov/Tadmor()), Abgrall(,).

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+ (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± 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).

6 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) C M ( t ttv + x xxv ). (3) High Order scheme Let S A denote a high order (possibly unstable) scheme: S A (u n )(x) := u n (x) th A (x, D k, u,..., D u n (x), D + u n (x),..., D k,+ u n (x)) (4) High order consistency: There exists k 2,and < l < k, for any v = v(t, x) of class C l+, there exists C A,l, E S A(v)(t, x) C A,l ( t l l+ t v + x l l+ x v ). (5)

7 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 (), and also obtain error estimates. This is the core of the present work. In our approach we adapt an idea of Froese and Oberman (3) (for second order HJ equations) to treat mainly the case of evolutive first order PDEs.

8 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 (), and also obtain error estimates. This is the core of the present work. In our approach we adapt an idea of Froese and Oberman (3) (for second order HJ equations) to treat mainly the case of evolutive first order PDEs.

9 filtered scheme: The scheme we propose is then u n+ j ( S F (ui n ) := S M (uj n ) + ɛ tf ) S A (u n j ) S M (u n j ) ɛ t (6) 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.

10 Filtered function Instead of the Froese and Oberman s filter function i.e. F (x) = sign(x) max( x, ), we used new filter function i.e. F (x) := x x : Froese and Oberman s filter function and new filter function. To keep high order when h A h M S ɛ i.e. A (uj n ) S M (uj n ) ɛ t SF S A Otherwise F = and S F = S M, i.e., the monotone scheme itself.

11 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 ( ) (7) 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 t, where C is constant and ɛ as t.

12 Convergence Theorem Theorem Assume (A)-(A2), and v bounded. We assume also that S M satisfies (A3)-(A5), and F. Let u n denote the filtered scheme (6). Let vj n := v(t n, x j ) where v is the exact solution of (). Assume < ɛ c x (8) for some constant c >. (i) The scheme u n satisfies the Crandall-Lions estimate u n v n C x, n =,..., N. (9) for some constant C independent of x.

13 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 (8)). Then it holds u n v n C x, n =,..., N, () for some constant C independent of x.

14 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 l k and v be a C l+ function on N. Assume that ( ) (C A, + C M ) v tt t + v xx x ɛ. () 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).

15 Tuning of the parameter ɛ Bounds for ɛ : Upper bound: ɛ C x, with constant C > ( to have error estimates and convergence ). Lower bound: 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.

16 Eikonal equation in D Example 2. { vt + v x =, t >, x ( 2, 2), v(, x) = v (x), := max(, 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-4.67 Table : L 2 errors for filtered scheme, Central finite difference (CFD) scheme, ENO (2nd order) scheme with RK2 in time.

17 .5 t=.5 t=.3.5 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 = 6 mesh points).

18 D steady equation Example 3. (as in Abgrall [6]) x := { vx = f (x), on (, ) v() = v() = f (x) := 3x 2 + a, a := 2x3 2x, Filter (ɛ = 5 x) CFD ENO M L error order L error order L error order.5e inf E E inf -.78E E inf E E inf E-3. Table : L Errors for filtered scheme, CFD scheme, and RK2-2nd order ENO scheme. Filter can stabilize an otherwise unstable scheme

19 Advection with obstacle Example 4. (as in Bokanowski et al. [9]) { min(vt + v x, g(x)) =, t >, x [, ], u(, x) =.5 + sin(πx) x [, ], 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-2.3.7E E E E E E E-5.98 Table : Local L errors for filtered scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time.

20 .5 t=.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.

21 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(, (x )2 y 2 ) 4, where. is the Euclidean norm and r =.5 with Drichlet boundary conditions. r 2 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-..57E E E E E E E E E E-3.77 Table : L 2 errors for filtered scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time.

22 y y t= t= Numerical front exact front y x x t=.6 3 t=.6 Numerical front exact front y x x

23 Eikonal in 2D Example 7. Ω = ( 3, 3) 2 { ( v(, x, y) = v (x, y) =.5.5 max max(, (x )2 y 2 r 2 ) 4, max(, (x+)2 y 2 ) ). 4 r 2 where. is the Euclidean norm and A ± := (±, ) 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-3.77 Table : L 2 errors for filtered scheme, Central finite difference (CFD) scheme, ENO (second order) scheme and RK2 in time.

24 y.5 t= 3 2 t= Numerical front exact front y x x t= t=.6 Numerical front exact front.5 y y x x

25 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.

26 Tanti Auguri

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28 Organizing team.