Generalized Clenshaw-Curtis quadrature method for systems of linear ODEs with constant coefficients

In this paper, we consider high precision numerical methods for the initial problem of systems of linear ordinary differential equations (ODEs) with constant coefficients. It is well-known that the analytic solution of such a system of linear ODEs involves a matrix exponential function and an integral whose integrand is the product of a matrix exponential and a vector-valued function. We mainly consider numerical quadrature methods for the integral term in the analytic solution and propose a generalized Clenshaw-Curtis (GCC) quadrature method. The proposed method is then applied to the initial-boundary value problem for a heat conduction equation and a Riesz space fractional diffusion equation, respectively. Numerical results are presented to demonstrate the effectiveness of the proposed method.