Numerical methods for nonlinear second-order hyperbolic partial differential equations. I. Time-linearized finite difference methods for 1-D problems

Abstract

A three-parameter family of finite difference methods for one-dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms, is presented and shown to result in two-time level semi-discrete equations. Second-order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three-time level linear finite difference equations which are second-order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1:2:1 to the past, current and future time levels. Second-order accurate in time, time-linearized, three-point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth-order accurate in space, are also presented, and their linear stability analyzed. An exhaustive comparison between analytical and numerical results indicates that the accuracy of time-linearized methods increases as the time step and grid spacing are decreased, and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure, and the best accuracy is achieved when the three implicitness parameters are equal to 1/2. It is also shown that the accuracy of time-linearized methods is a strong function of the wave speed; as the wave speed is increased and the wave equation tends to a parabolic one, the methods are still three-time level techniques which may not preserve the positivity of the solution, whereas a Crank–Nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions. It is also shown that, for one-dimensional wave equations with monotonic kinetics and time-dependent forcing, there are very few differences between second- and first-order time-accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small, and the discretization of the reaction terms plays a less important rule than that of the diffusion terms.