NUMERICAL APPROXIMATION AND ANALYSIS FOR PARTIAL INTEGRODIFFERENTIAL EQUATIONS OF HYPERBOLIC TYPE

Abstract

We consider the numerical analysis for a partial integrodifferential equation of hyperbolic type. The central difference formula and the second-order convolution quadrature are applied to construct the numerical scheme, where the convolution quadrature method could accommodate the complicated case that the explicit form of the memory kernel is not available. We propose a novel analysis to prove the finite-time stability of the numerical solutions and specify the condition that ensures the long time stability. We also prove error estimates for the numerical scheme based on a newly developed approximate result of convolution quadrature for convolution of nonsmooth functions. Finally, we extend the developed methods to construct and analyze a numerical scheme for the corresponding nonlinear problems. Numerical experiments are performed to substantiate the theoretical findings.