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A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial derivatives are approximated by fourth-order finitedifference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional heat equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two-and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions.

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SUMMARYIn this paper numerical methods for solving ÿrst-order hyperbolic partial di erential equations are developed. These methods are developed by approximating the ÿrst-order spatial derivative by third-order ÿnite-di erence approximations and a matrix exponential function by a third-order rational approximation having distinct real poles. Then parallel algorithms are developed and tested on a sequential computer for an advection equation with constant coe cient and a non-linear problem.