An adaptive integrating factor midpoint method for second order evolution equations

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.