Stiffness and Accuracy in the Method of Lines Integration of Partial Differential Equations

Abstract

The method of lines (MOL) is a flexible algorithm for the numerical integration of systems of simultaneous nonlinear partial differential equations (PDE’s) with an initial value independent variable. Our experience has indicated that it can be applied to a spectrum of scientific and engineering problems and therefore we have for several years undertaken a developmental effort to implement the MOL in general FORTRAN-based programs for PDE’s. Basically, the spatial derivatives (i.e., the derivatives with respect to the boundary value independent variable (s)) are replaced with finite difference approximations. The resulting system of initial value ordinary differential equations (ODE’s) is then integrated numerically to obtain a solution along lines of constant spatial independent variable(s). Multidimensional PDS’s (i.e., with more than one spatial variable) can be handled as well as simultaneous PDE’s with differing numbers of spatial independent variables. Since the MOL is oriented toward initial value ODE’s, it is particularly well suited for the numerical integration of systems of mixed ODE/PDE’S. These mixed systems occur in physical problems in which the ODE’s are boundary conditions for the PDE’s.