Efficient Time Integration Methods for Semilinear Damped Wave Equations

This thesis develops efficient time integration methods for semilinear wave equations with damping and multiscale features. In particular, we study two promising schemes that exploit the underlying structure of the equations: implicit–explicit (IMEX) schemes and exponential integrators.

We analyze an IMEX scheme for semilinear wave equations with strong damping, prove unconditional stability and derive error bounds for the fully discrete problem in a general spatial discretization framework. As an application, we provide convergence results for the heterogeneous multiscale method applied to wave equations with highly oscillatory coefficients.

In addition, we study exponential integrators for semilinear damped wave equations, which require the efficient approximation of matrix functions. We present a Chebyshev-based method for approximating these matrix functions, which relies on a Faber expansion over suitably chosen ellipses and includes both a residual error estimator and a novel, efficient strategy for selecting the ellipse parameters.

Numerical experiments demonstrate the accuracy and efficiency of the proposed methods.