An adaptive integrating factor midpoint method for second order evolution equations

TL;DR

This paper develops an adaptive integrating factor midpoint method for second order wave equations, providing optimal error estimates and an efficient adaptive time-stepping strategy verified by numerical experiments.

Contribution

It introduces a new adaptive method with optimal a posteriori error estimates for wave equations, improving accuracy and efficiency over existing techniques.

Findings

Optimal order a posteriori error estimates derived

Adaptive algorithm demonstrates high efficiency in numerical tests

Error estimator's convergence rate verified numerically

Abstract

In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear approximate solutions, and derive suboptimal order residual-based error estimates using the energy technique. Hence the key is introducing a continuous, piecewise quadratic time reconstruction to establish optimal order error bounds. Based on the reliable a posteriori error control, we develop an adaptive time-stepping strategy. Numerical examples are implemented to verify the convergence rate of an error estimator and the high efficiency of the adaptive algorithm.