An Algebraic Approximation for the Matrix Exponential in Singularly Perturbed Boundary Value Problems

Abstract

A symmetric difference scheme for linear, stiff, or singularly perturbed boundary value problems of first order with constant coefficients is constructed, being based on a stability function containing a matrix square root. Its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix. Local damping of errors, uniform stability, and uniform second-order convergence are proved. The computation of the specific matrix square root by a well-known, stable variant of Newton’s method is discussed. A numerical example confirming the results is given.