Abstract
We construct and analyze a second-order implicit–explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate.
Highlights
In this paper we construct and analyze an implicit–explicit (IMEX) time integration scheme for second-order semilinear wave equations of the form u (t) + Bu (t) + Au(t) = f (t, u(t))in a suitable Hilbert space
The main contribution of this paper is to provide a full discretization error analysis of semilinear wave equations in a quite general framework
The paper is organized as follows: in Sect. 2 we present the problem setting, introduce the IMEX scheme for second-order wave equations, and state a second-order error bound for the time discretization error
Summary
We show that the scheme is unconditionally stable in the sense that the time-step size is only restricted by the Lipschitz constant of f but not by the linear operators A and B We combine this IMEX scheme with an abstract, nonconforming space discretization within the framework of [11,12,13]. The main contribution of this paper is to provide a full discretization error analysis of semilinear wave equations in a quite general framework Such an error analysis does not even exist for the Crank–Nicolson scheme, which is covered as a byproduct of our analysis of the IMEX scheme. The challenge of such a rigorous analysis is that it applies to abstract, non-conforming space discretizations of semilinear wave-type equations. We first introduce the problem setting and present the IMEX scheme and its properties
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