Evolution equations with spectral methods: the case of the wave equation

Transcription

1 Evolution equations with spectral methods: the case of the wave equation Laboratoire de l Univers et de ses Théories (LUTH) CNRS / Observatoire de Paris, France in collaboration with Silvano Bonazzola November,

2 Plan 1 and spectral methods 2 3 Absorbing boundary General form of the solution Absorbing BC for l 2

3 It seems that, in general, there is no efficient spectral decomposition for the time coordinate... use of finite-differences! t is discretized (usually) on an equally-spaced grid, with a times-step δt : U J = U(J δt). du dt = F (U) = L(U) + Q(U) Study, for different integration of : stability : n U n Ce Kt U 0, for some δt < δ lim, region of absolute stability : when considering du dt = λu, the region in the complex plane for λδt for which U n is bounded for all n, unconditional stability : if δ is independent from N (level of spectral truncation).

4 One-dimensional study To use the knowledge of the region of absolute stability, it is necessary to diagonalize the matrix L and study its eigen-values λ i. In one dimension : First-order Fourier For L = d/dx, one finds max λ i = O (N) First-order Chebyshev For L = d/dx, one finds max λ i = O ( N 2) Second-order Fourier For L = d/dx 2, one finds max λ i = O ( N 2) Second-order Chebyshev For L = d 2 /dx 2, one finds max λ i = O ( N 4)

5 integration Most popular... Explicit first-order Adams-Bashford scheme (a.k.a forward Euler) : U n+1 = U n + δtf (U n ), second-order Adams-Bashford scheme : [ 23 U n+1 = U n + δt 12 F (U n ) F ( U n 1) F ( U n 2) ], Runge-Kutta... All these exhibit a bounded region of absolute stability K > 0, δt K/ max λ i (Courant condition...).

6 integration Most popular... Implicit Adams-Moulton : first-order (a.k.a backward Euler scheme) U n+1 = U n + δtf ( U n+1), second-order (a.k.a. Crank-Nicholson scheme) U n+1 = U n δt [ F ( U n+1) + F (U n ) ]. Both have an unbounded region of absolute stability in the left complex half-plane unconditionally stable. Higher-order AM have only a bounded region of absolute stability. Schemes can be mixed and various source terms can be treated in different ways (e.g. linear implicit / non-linear explicit).

7 The three-dimensional wave equation in spherical coordinates : with φ = 1 2 φ c 2 t φ r φ r r + 1 r 2 θϕφ = σ; θϕ 2 θ tan θ θ + 1 sin 2 θ 2 ϕ 2 In 1D, it admits two characteristics : ±c : f(ct x) and f(ct + x). To be well-posed, the initial-boundary value problem needs : φ(t = 0) and φ/ t(t = 0), a boundary condition at every domain boundary (Dirichlet, von Neumann, mixed).

8 An explicit scheme for the wave equation Using a second-order scheme to evaluate the second time derivative 2 φ t 2 = φj+1 2φ J + φ J 1 δt 2 + O ( δt 4), t=t J one recovers the forward Euler scheme φ J+1 = 2φ J φ J 1 + δt 2 ( φ J + σ ) + O(δt 4 ). Solution of the initial-boundary value problem inside a sphere or r R : initial profiles at t = t 0 and t = t 1, t > t 1, a value for φ(r = R). With spectral methods using Chebyshev polynomials in r, time-step limitation is coming from the second radial derivative : δt 2 K/N 4. Complete 3D problem regularity at the origin too, for l > 1.

9 With the same formula for the second time derivative and the Crank-Nicholson scheme : ] ( ) [1 δt φ J+1 = 2φ J φ J 1 + δt 2 φj 1 + σ J. One must invert the operator 1 1/2δt 2 ; one way is : consider the spectral representation of φ in terms of spherical harmonics ( θϕ Yl m = l(l + 1)Yl m ) ; solve the ordinary differential equation in r as a simple linear system, using e.g. the tau method. one can add boundary and regularity depending on the multipolar momentum l. beware of the condition number of the operator matrix! sometimes regularity is better imposed (stable) using a Galerkin base.

10 Domain boundaries Contrary to the Laplace operator, the d Alembert one is not invariant under inversion / sphere. one cannot a priori use a change of variable u = 1/r! the distance between two neighboring grid points becomes larger than the wavelength... domain of integration bounded (e.g. within a sphere of radius R). Two types of BCs : reflecting BC : φ(r = R) = 0, absorbing BC...

11 An absorbing BC can be seen in 1D : at x = 1 one imposes no incoming characteristic only f(ct x) mode. In spherical 3D geometry : asymptotically, the solution must match equivalently, 1 φ r f(ct r), r (rφ) lim + c (rφ) = 0. r t t At finite distance R : ( 1 φ c t + φ r + φ ) = 0; r r=r which is exact in spherical symmetry.

12 General form of the solution The homogeneous wave equation φ = 0 admits as asymptotic development of its solution f k (t r, θ, ϕ) φ(t, r, θ, ϕ) = r k. k=1 One can show that the contribution from a mode l exists only for k l + 1. Moreover, the operators : B 1 f = f t + f r + f ( r, B n+1f = t + r + 2n + 1 ) B n f r are such that the condition B n φ = 0 matches the first n terms (B n φ = O ( 1/r 2n+1) ). It follows that B n φ = 0 is a n th -order BC, is exact for all modes l n 1, is asymptotically exact with an error decreasing like 1/R n+1, is the generalization of the at finite distance (n = 1).

13 Absorbing BC for l 2 The condition B 3 φ = 0 at r = R writes ( (t, θ, ϕ), B 1 φ r=r = t + r + 1 ) φ(t, r, θ, ϕ) r = ξ(t, θ, ϕ), r=r with ξ(t, θ, ϕ) verifying a wave-like equation on the sphere r = R 2 ξ t 2 3 4R 2 θϕξ + 3 ξ R t + 3ξ 2R 2 = 1 ( φ 2R 2 θϕ R φ ) r. r=r easy to solve if ξ is decomposed on the spectral base of spherical harmonics! looks like a perturbation of the... exact for l 2 and the error decreases as 1/R 4 for other modes.