Numerical quadrature methods for systems of linear ODEs with constant coefficients

Abstract 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 (AS) of the initial value problem for a system of linear ODEs with constant coefficients involves a matrix exponential function, and an integral whose inte-grand is the product of a matrix exponential function and a vector-valued function. We mainly consider numerical quadrature methods for the integral term in the analytic solution. To simplify the discussion, we consider the case where the coefficient matrix is real and symmetric, and compute the corresponding matrix exponential function by using matrix spectral factorization. We propose a product quadrature (PQ) method, including its detailed computational procedure and its convergence analysis. Then the proposed methods are applied to the initial-boundary value problem for a heat conduction equation and a Riesz space fractional diffusion equation, respectively. Applying the central finite difference operator to discretize the second-order space derivative or a weighted shifted Grünwald difference (WSGD) operator to discretize the Riesz space fractional derivative, the initial-boundary value problem for the above mentioned equations is transformed to the initial value problem of a system of linear ODEs with constant coefficients. We then transform the systems of linear ODEs to symmetric ones if necessary. Numerical results presented to demonstrate the effectiveness of the proposed methods. Mathematics Subject Classification (2010) MSC 65M06 · MSC 26A33 · MSC 65M12