On averaged exponential integrators for semilinear wave equations with solutions of low-regularity

Abstract

In this paper we introduce a class of second-order exponential schemes for the time integration of semilinear wave equations. They are constructed such that the established error bounds only depend on quantities obtained from a well-posedness result of a classical solution. To compensate missing regularity of the solution the proofs become considerably more involved compared to a standard error analysis. Key tools are appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas. We include numerical examples to illustrate the advantage of the proposed methods.

A second-order low-regularity correction of Lie splitting for the semilinear Klein--Gordon equation

“…(Ω) for spatial dimension d = 1, 2, 3; see Theorem 3.1. In the one-dimension case, the proposed method is shown to have a convergence order arbitrarily close to 5 3 in the energy space H 1 (Ω) × L 2 (Ω) for solutions in the same space, i.e. no additional regularity in the solution is required.…”

“…The numerical approximation of semilinear Klein-Gordon equations in the form of (1.1) has been extensively studied in computational mathematics. A large variety of numerical schemes for approximating the time dynamics of the semilinear Klein-Gordon equation has been proposed and analyzed, including trigonometric/exponential integrators that are based on the variation-of-constants formula (for example, see [5,11,13,17,29]), splitting methods (for example, see [1,2,5,10]), finite difference methods (such as the Crank-Nicolson and Runge-Kutta methods, see [6,16,19,[21][22][23]26]), and symplectic methods [7,8,14].…”

“…The analyses in these articles (for example in [5,11,17,29]) have shown that for initial data (u 0 , v 0 ) in the physically natural energy space H 1 (Ω) × L 2 (Ω), so that the solution (u, ∂ t u) is bounded in the energy space H 1 (Ω)×L 2 (Ω) uniformly for t ∈ [0, T ], the classical time-stepping methods such as splitting methods, Runge-Kutta methods, trigonometric integrators, and averaged exponential integrators, can approximate the solution (u, ∂ t u) with second-order convergence in the weaker space L 2 (Ω) × H −1 (Ω), but only with first-order convergence in the energy space H 1 (Ω) × L 2 (Ω) itself. Moreover, the second-order approximation to (u, ∂ t u) in the energy space H 1 (Ω) × L 2 (Ω) generally requires the initial data to be in the stronger space H 2 (Ω) × H 1 (Ω).…”

Exponential Integrators for Quasilinear Wave-Type Equations

“…As a consequence, we only consider first-and second-order methods and analyze the convergence in the appropriate spaces. This approach was already used by the authors in [3].…”

Discrete conservation laws for finite element discretisations of multisymplectic PDEs

“…A backward error analysis for multi-symplectic schemes in the PDE setting is not yet fully developed, [49,35]. On the other hand, being able to bound the approximation by the energy often proves to be an invaluable property for convergence studies of both geometric numerical integrators [23,13,12] and finite element schemes via energy arguments [57].…”