Nonstandard finite difference equations for ODEs and 1-D PDEs based on piecewise linearization

Abstract

A method for the solution of initial and boundary value problems in nonlinear, ordinary differential equations, and for one-dimensional, partial differential equations which provides C 1 solutions is presented. The method is based on the linearization of the differential equation in intervals which contain only two grid points and provides three-point, nonstandard finite difference equations for the nodal amplitudes. The method is applied to steady reaction-diffusion equations, two-point, singularly perturbed boundary value problems and the steady Burgers equation, and compared with standard finite difference and finite element formulations. For one-dimensional, partial differential equations, the temporal derivatives are first discretized, and the resulting ordinary differential equation accounts for both the temporal and spatial stiffnesses and is solved by means of piecewise linearization. Since the linearization includes a Jacobian matrix, it may be easily employed to refine the mesh where steep gradients occur.