Linear algebra for exponential integrators

Transcription

1 Linear algebra for exponential integrators Antti Koskela KTH Royal Institute of Technology Beräkningsmatematikcirkus, 28 May 2014

2 The problem Considered: time integration of stiff semilinear initial value problems u (t) = Au(t) + g(t, u(t)), u(t 0 ) = u 0, where u(t) R n, A R n n (n large), g a nonlinear function with a moderate Lipschitz constant. Specifically considered: problems arising from the spatial semidiscretization of partial differential equations. E.g.: A comes from the discretization of the linear part of a PDE, using finite elements or finite differences. Antti Koskela, KTH Linear algebra for exponential integrators 2

3 Example Consider the Brusselator problem in 2D, u t = ν u + µu u + av (b + 1)u + u2 v, v t = ν v + µv v + bu u2 v, where A and B constants. Write as w t = Aw + g(w), where [ ] ν + µu + (b + 1) a A =, w = b ν + µv [ ] u. v Antti Koskela, KTH Linear algebra for exponential integrators 3

4 Example Consider the advection-diffusion equation t y(t, x) = ɛ xx y(t, x) + α x y(t, x), y(0, x) = 16(x(1 x)) 2, x [0, 1] subject to homogenous Dirichlet boundary conditions. Perform spatial discretization using the central differences ODE: u (t) = Au(t), u(0) = u 0. Compute the Krylov approximation of: 1) exp(ha)u 0 2) (I + h 2 A)(I h 2 A) 1 u 0 Antti Koskela, KTH Linear algebra for exponential integrators 4

5 Krylov subspace The Arnoldi iteration gives an orthogonal basis Q k R n k for the Krylov subspace K k (A, b) = span{b, Ab, A 2 b,..., A k 1 b}, and the Hessenberg matrix H k = Q T k AQ k R k k. Antti Koskela, KTH Linear algebra for exponential integrators 5

6 Approximation of matrix functions Cauchy s integral formula: for any analytic function f defined on D C, it holds f (A) = 1 f (λ)(λi A) 1 dλ, 2πi C where C is a closed curve inside the domain D enclosing σ(a). By using the approximation (λi A) 1 b Q k (λi H k ) 1 Q T k b, and letting C enclose the field of values F(A), the approximation f (A)b Q k f (H k )Q T k b is obtained. Antti Koskela, KTH Linear algebra for exponential integrators 6

7 Example 10 4 relative 2 norm error Arnoldi error for the Crank-Nicholson Arnoldi error for the matrix exponential Number of matrix vector multiplications Figure: Convergence of different Krylov approximations. Here: h = , ha 199. Antti Koskela, KTH Linear algebra for exponential integrators 7

8 Exponential integrators Exponential integrators are based on the variation-of-constants formula u(t) = e (t t 0)A u 0 + t t 0 e (t τ)a g(τ, u(τ)) dτ. which gives the the exact solution at time t. Example: exponential Euler method u 1 = e ha u 0 + hϕ 1 (ha)g(t 0, u 0 ) where h denotes the step size and ϕ 1 is the entire function ϕ 1 (z) = e z 1. z Antti Koskela, KTH Linear algebra for exponential integrators 8

9 Fast computation of series of ϕ - functions Computation of one time step can be done using the following lemma (Al-Mohy and Higham 2010). Lemma Let A R d d, W = [w 1, w 2,..., w p ] R d p, h R and à = [ ] A W R (d+p) (d+p), J = 0 J with e 1 = [1, 0,..., 0] T. Then it holds e ha u 0 + [ ] 0 0, v I p = p h k ϕ k (ha)w k = [ I d 0 ] e hãv 0. k=1 [ u0 e 1 ] R d+p Antti Koskela, KTH Linear algebra for exponential integrators 9

10 References A. Koskela and A. Ostermann. Exponential Taylor methods: analysis and implementation. Computers and Mathematics with Applications (2013). A. Koskela and A. Ostermann. A moment-matching Arnoldi iteration for linear combinations of ϕ-functions. (submitted for publication). A. Koksela. Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method. To appear in ENUMATH proceedings. Antti Koskela, KTH Linear algebra for exponential integrators 10