Numerical schemes for short wave long wave interaction equations

Transcription

1 Numerical schemes for short wave long wave interaction equations Paulo Amorim Mário Figueira CMAF - Université de Lisbonne LJLL - Séminaire Fluides Compréssibles, 29 novembre 21 Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 1 / 38

2 Outline A short wave long wave interaction system: Convergence of semidiscrete finite difference schemes A Schrödinger Conservation law system: Convergence of semidiscrete finite volume type schemes Numerical Results Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 2 / 38

3 Short wave long wave interaction equations Systems of coupled nonlinear PDEs arising in gravity-capillary waves, plasma physics, etc... (Benney, 1977) { i t u + ic 1 x u + xx u = αu v + γ u 2 u t v + c 2 x v + µ 3 x v + ν x v 2 = β x ( u 2 ), t [, ), x R. Nonlinear Schrödinger equation coupled with: Dispersive equations (KdV) Hyperbolic equations (scalar conservation laws or systems) Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 3 / 38

4 Approximation of a short wave long wave interaction system A short wave long wave interaction system { i t u + xx u = α u 2 u + vu t v = x ( u 2 ). Tsutsumi & Hatano (1994): Global well-posedness of the Cauchy problem in H j+1/2 (R) H j (R), j 1. Bekiranov, Ogawa, Ponce (1998): Local well-posedness of the Cauchy problem in H s (R) H s 1/2 (R), s. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 4 / 38

5 Approximation of a short wave long wave interaction system u h (t) = (u j (t)) j Z, v h (t) = (v j (t)) j Z, h = x j+1 x j u j (t), v j (t) u(x j, t), v(x j, t) A semidiscrete finite difference scheme i du j dt + h u j = α u j 2 u j + v j u j dv j dt = D ( u j 2 ) u j () = u j, v j () = v j, h u j := 1 h 2 (u j+1 2u j + u j 1 ), D u j := 1 2h (u j+1 u j 1 ). P h 1u h := Piecewise linear, continuous interpolator Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 5 / 38

6 Convergence result Theorem (PA, M.Figueira, C.R.A.S. 29) Let (u, v ) H 1 (R) L 2 (R). For each T >, the sequences u h, v h satisfy: P h 1u h u in L ([ T, T ]; H 1 (R)) weak P h 1u h u in L ([ T, T ]; L 2 loc (R)) P h 1v h v in L ([ T, T ]; L 2 (R)) weak with (u, v) the unique strong solution, (u, v) ( C([ T, T ]; L 2 ) L ([ T, T ]; H 1 ) ) C([ T, T ]; L 2 ). Moreover, if (u, v ) H 2 (R) H 1 (R), then P h 1u h u in L ([ T, T ]; H 1 loc (R)) P h 1v h v in L ([ T, T ]; L 2 loc (R)). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 6 / 38

7 Sketch of the proof The theorem follows from the following estimates: For all h > there exist positive continuous functions a(t), b(t), c(t) such that D + u h (t) L 2 a(t) (bound on H 1 norm of u h ) h u h (t) L 2 b(t) (bound on H 2 norm of u h ) v h L 2 c(t) (bound on L 2 norm of v h ). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 7 / 38

8 Sketch of the proof Energy conservation: d dt + Conservation of u h L 2 { 1 2 D +u h 2 L 2 + α 4 uh 4 L v, uh 2 L 2 + Discrete Gagliardo Niremberg Sobolev inequalities: } = φ L C φ 1/2 D L 2 + φ 1/2, φ L 2 L 4 C φ 3/4 D L 2 + φ 1/4 L 2 D + u h 2 L 2 c + c v h L 2 D + u h 1/2 L 2. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 8 / 38

9 D + u h 2 L 2 c + c v h L 2 D + u h 1/2 L 2. But: Bound on v h L 2: 1 d 2 dt v h 2 L = D 2 u h 2, v h L 2 v h L 2 c + c D + u h 2 L 2 c + c D + u h 1/2 L 2 t (1 + D + u h L 2) 3/2 c + c ( t + c D + u h (s) 3/2 ds L 2 t Gronwall lemma D + u h (s) L 2 a(t). ) D + u h (s) 3/2 ds D L 2 + u h 1/2. L 2 (1 + D + u h (s) L 2) 3/2 ds Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 9 / 38

10 A Schrödinger Conservation law system (Dias, Frid, Figueira, A.R.M.A. 21) Schrödinger Conservation law system { i t u + xx u = u 2 u + αg(v)u t v + x f (v) = α x (g (v) u 2 ) t [, ), x R. Nonlinear conservation law with flux f Coupling function g g has compact support (preserves physical domains) Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 1 / 38

11 Scalar conservation laws: entropy solutions Background on scalar conservation laws v(x, t) : R [, ) R t v + x f (v) =, v(x, ) = v (x) L (R) Smooth solutions develop discontinuities in finite time Non-uniqueness of weak solution Convex entropy η(v), entropy flux: q (v) = η (v)f (v) Entropy inequalities: t η(v) + x q(v) in D. There is a unique weak solution verifying the entropy inequalities: the entropy solution. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 11 / 38

12 Entropy solutions Entropy solution of { i t u + xxu = u 2 u + g(v)u t v + x f (v) = x (g (v) u 2 ) u(, t) H 1 (R) solution of i t u + xx u = u 2 u + g(v)u in D v L (R [, )) verifies t η(v) + x (q 1 (v) q 2 (v) u 2 ) (η (v)g (v) q 2 (v)) x u 2 in D. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 12 / 38

13 A class of Semidiscrete schemes We consider a class of semidiscrete schemes inspired by finite volume schemes: u j (t) xj+1/2 xj+1/2 u(x, t)dx, v j (t) v(x, t)dx, j Z, t >. x j 1/2 x j 1/2 i t u j + 1 h 2 (u j+1 2u j + u j 1 ) = u j 2 u j + g(v j )u j t v j + 1 h ( Hj,+ (v j, v j+1, u j 2, u j+1 2 ) + H j, (v j, v j 1, u j 2, u j 1 2 ) ) = Finite volume scheme Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 13 / 38

14 Finite volume schemes motivation h = x j+1/2 x j 1/2 t v + x f (v) = 1 h 1 xj+1/2 t v dx t v j h 1 h x j 1/2 xj+1/2 x j 1/2 xj+1/2 x j 1/2 dx x f (v) dx = 1 ( f (v(xj+1/2, t)) f (v(x h j 1/2, t)) ) 1 ( f+ (v j, v j+1 ) + f (v j, v j 1 ) ) h Numerical flux functions f ± Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 14 / 38

15 Finite volume methods The numerical flux functions f ± (v 1, v 2 ) must be Monotone: 1 f ± (v 1, v 2 ), 2 f ± (v 1, v 2 ). Conservative: f + (v 1, v 2 ) = f (v 2, v 1 ). Consistent with the flux function: f ± (v, v) = ±f (v). H j,+ (v 1, v 2, a 1, a 2 ) are numerical flux functions consistent with f (v) g (v) u 2 : H j,+ (v, v, u 2, u 2 ) = f (v) g (v) u 2. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 15 / 38

16 Examples A Lax Friedrichs scheme H j,± (v, w, a, b) = ± 1 ( ) 1 f (v) + f (w) (w v) 2 2λ ± ( g (v) a 2 g (w) b 2) 1 2γ (w v)1 (a + b) 2 A Godunov scheme H j,± (v 1, v 2, a 1, a 2 ) = min v 1 s v 2 ± ( f (s) g (s) 1 2 (a 1 + a 2 ) ), v 1 v 2 max v 2 s v 1 ± ( f (s) g (s) 1 2 (a 1 + a 2 ) ), v 2 v 1 Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 16 / 38

17 Main convergence result i t u j + 1 h 2 (u j+1 2u j + u j 1 ) = u j 2 u j + g(v j )u j t v j + 1 h ( Hj,+ (v j, v j+1, u j 2, u j+1 2 ) + H j, (v j, v j 1, u j 2, u j 1 2 ) ) = Theorem (PA, M.Figueira, submitted.) Let (u h, v h ) be defined by the semidiscrete scheme above. Then there exist functions u C([, ); H 1 (R)), v L (R [, )), entropy solutions of the Cauchy problem such that, up to a subsequence, (u h, v h ) converge to (u, v) in L 1 loc (R [, )). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 17 / 38

18 Convergence framework: Compensated Compactness If t η(v h ) + x ( q1 (v h ) u h 2 q 2 (v h ) ) { compact of W 1,2 loc }, then (v h ) h> is compact in L 1 loc (R [, + )) and if (u h ) h> is bounded in (discrete) H 1 (R) (u h, v h ) (u, v) solutions of the Schrödinger Conservation law system. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 18 / 38

19 Estimates u h (t) 2 = C (conservation of L 2 norm) v h (t) M (uniform L bound) From the conservation law we get the essential viscosity estimate: t j Z vj+1 η (v) { f (v) u j 2 g (v) H(v j, v j+1, u j 2, u j 2 } ) dv ds v j }{{} c + c t D + u h (s) 2 ds numerical flux function Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 19 / 38

20 Next: from the equations we get the energy estimate 1 2 D +u h uh g(v j), u j 2 L 2 = c + t g (v j ) t v j, u j 2 L 2ds. Using the scheme and the viscosity estimate gives t g (v j ) t v j, u j 2 L 2ds t ) t sup D + u h 2 (c + c D + u h 2 ds + c D + u h 2 ds. (,t) Gronwall D + u h (t) 2 a(t) (H 1 estimate of u h ). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 2 / 38

21 The viscosity estimate now becomes t vj+1 η (v) { f (v) u j 2 g (v) H(v j, v j+1, u j 2, u j 2 } ) dv ds j Z v j }{{} numerical flux function c(t). In the case of the Lax Friedrichs scheme, one has the simpler quadratic total variation estimate: k > such that k t j Z which is false in general. (1 + uj h 2 )(vj+1 h vj h ) 2 ds c(t) Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 21 / 38

22 So: do we have t η(v h ) + x ( q1 (v h ) u h 2 q 2 (v h ) ) { compact of W 1,2 loc }? If φ is a test-function, we have t η(v h ) + x (q 1 (v h ) q 2 (v h ) u h 2 ), φ D D = h φ j η ( (v j ) t v j dt φ j+1 q1 (v j+1 ) q 1 (v j ) ) dt j Z + j Z j Z ( φ j+1 q2 (v j+1 ) u j+1 2 q 2 (v j ) u j 2) dt + use the scheme to replace t v j : t v j = 1 ( Hj,+ (v j, v j+1, u j 2, u j+1 2 ) + H j, (v j, v j 1, u j 2, u j 1 2 ) ) h Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 22 / 38

23 The viscosity estimate t j Z vj+1 v j c(t) η (v) { f (v) u j 2 g (v) H(v j, v j+1, u j 2, u j 2 ) } dv ds allows us to bound the resulting terms. This gives t η(v h ) + x ( q1 (v h ) u h 2 q 2 (v h ) ) { compact of W 1,2 loc } and thus compactness of v h. Using similar techniques, one then proves that the resulting limit pair (u, v) is an entropy solution of the problem. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 23 / 38

24 Numerical sheme Semi-implicit Cranck Nicholson scheme for the Schrödinger equation i 1 τ (un+1 j uj n ) + 1 ( u n+1 2h 2 j+1 + un j+1 2(u n+1 j + uj n ) + u n+1 j 1 + ) un j 1 = 1 2 (un+1 j + uj n ) (un+1 j + uj n ) + g(vj n ) 1 2 (un+1 j + uj n ), Semi-implicit Lax Friedrichs scheme for the conservation law 1 n+1 (vj vj n ) = 1 n (f (v τ 2h j+1) f (vj 1)) n + 1 2h (g (vj+1) u n j+1 n 2 g (vj 1) u n j 1 n 2 ) + 1 n+1 n+1 (vj+1 2vj + v n+1 2λh + 1 2γh j 1 ) [ (v n+1 j+1 v n+1 j ) 1 2 ( un j 2 + u n j+1 2 ) (v n+1 j v n+1 j 1 )1 2 ( un j 2 + u n j 1 2 ) ]. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 24 / 38

25 Numerical results (I): a test-case { i t u + xx u = αvu t v = x ( u 2 ) Exact solutions: Traveling waves u(x, t) = e itλ e icx/2 C sech 2 (C(x ct)), v(x, t) = a sech 2 (C(x ct)). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 25 / 38

26 Numerical results (I): a test-case Initial Schrodinger Initial Conservation law Schrodinger solution Conservation law solution Initial data and numerical solution for t = 4. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 26 / 38

27 Numerical results (II): Full system, linear flux { i t u + xx u = u 2 u + vu t v + γ x v = x ( u 2 ) Exact solutions: Traveling waves u(x, t) = e itλ e icx/2 C sech 2 (C(x ct)), v(x, t) = a sech 2 (C(x ct)). Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 27 / 38

28 Numerical results (II): Full system, linear flux 1.8 Schrodinger Initial data Conservation law Initial data Schrodinger solution Conservation law solution Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 28 / 38

29 Numerical results (II): Full system, linear flux.1 Square of L2 error, linear flux, dt=.1, T = 3 u error v error.1 2 log(l2 Error) Number of points L 2 error between the exact solution and the computed solution, linear flux. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 29 / 38

30 Numerical results (III): Full system, nonlinear flux { i t u + xx u = vu t v + x v 2 = x ( u 2 ) Exact solutions: Standing waves (u, v) = (e ibt r(x), r(x)), r(x) = b(3/2) sech 2 ( bx/2) Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 3 / 38

31 Numerical results (III): Full system, nonlinear flux.1 Square of L2 error, Nonlinear flux, dt=.1, T = 3 u error v error.1 2 log(l2 Error) Number of points L 2 error between the exact solution and the computed solution, nonlinear flux. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 31 / 38

32 Numerical results (IV): Full system { i t u + xx u = u 2 u + g(v)u t v + x f (v) = x (g (v) u 2 ) Initial data: u (x) = e 5ix/2 6 sech( 3x), v (x) = χ [ 1,1] on the spatial domain [ 5, 5]. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 32 / 38

33 Numerical results (IV): Full system Full system, 5 pts, dx=.1, T = Full system, 5 pts, dx=.1, T = Schrodinger Initial data Conservation law Initial data Schrodinger Solution Conservation law Solution Full system, 5 pts, dx=.1, T = 1 Full system, 5 pts, dx=.1, T = Schrodinger Solution Conservation law Solution 1.4 Schrodinger Solution Conservation law Solution Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 33 / 38

34 Numerical results (IV): Full system Full system, 5 pts, dx=.1, T = 2 Full system, 5 pts, dx=.1, T = Schrodinger Solution Conservation law Solution 1.4 Schrodinger Solution Conservation law Solution Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 34 / 38

35 Numerical results (IV): Full system.8.6 Full system, 5 pts, dx=.1, T = 2.5 real part of u absolute value of u imaginary part of u Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 35 / 38

36 Summary We have considered a system of short wave long wave interaction equations and a Schrödinger Conservation law system. For both systems we have proved convergence results for semidiscrete schemes. The convergence is strong for the Schrödinger Conservation law system. We used Compensated Compactness to deal with the nonlinear equations. The proof of convergence uses mostly tools from hyperbolic conservation laws but relies heavily on the interaction between the two equations. The numerical implementation of the algorithm was successfully tested against known exact solutions and shows formation of new waves due to the interaction terms. Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 36 / 38

37 Open questions The fundamental uniform bound v h (t) M needs supp g compact (g is the coupling function). We have not observed explosion of v h (t) if supp g is not compact. Is there a hidden maximum principle for the Schrödinger Burgers system (i.e., g(v) v)? Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 37 / 38

38 Thank you for your attention! pamorim/ Paulo Amorim (CMAF - U. Lisboa) SW LW interactions 38 / 38