A new direct time integration method for the semi-discrete parabolic equations

Abstract

A direct time integration method is presented for the solution of systems of first order ordinary differential equations, which represent semi-discrete diffusion equations. The proposed method is based on the principle of the analog equation, which converts the N coupled equations into a set of N single term uncoupled first order ordinary differential equations under fictitious sources. The solution is obtained from the integral representation of the solution of the substitute single term equations. The stability and convergence of the numerical scheme is proved. The method is simple to implement. It is self-starting, unconditionally stable, accurate, while it does not exhibit numerical damping. The stability does not demand symmetrical and positive definite coefficient matrices. This is an important advantage, since the scheme can solve semi-discrete diffusion equations resulting from methods that do not produce symmetrical matrices, e.g. the boundary element method. The method applies also to equations with variable coefficients as well as to nonlinear ones. It performs well when long time durations are considered and it can be used as a practical method for integration of stiff parabolic equations in cases where widely used methods may fail. Numerical examples, including linear as well as non linear systems, are treated by the proposed method and its efficiency and accuracy are demonstrated.