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

摘要

A three-parameter family of finite difference methods for one-dimensional hyperbolic equations with damping and non-linearities 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 in-compatibilities 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. (c) 2007 Elsevier Inc. All rights reserved.