The large discretization step method for time-dependent partial differential equations

Abstract

A new method for the acceleration of linear and nonlinear time-dependent calculations is presented. It is based on the large discretization step (LDS, in short) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time-dependent differential operator. These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time-dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes. The efficiency of the LDS algorithms is defined as the cost of computing the fine grid solution relative to the cost of obtaining the same accuracy with the LDS methods. The LDS method’s typical efficiency is 16 for two-dimensional problems and 28 for three-dimensional problems for both linear and nonlinear equations. For a particularly good discretization of a linear equation, an efficiency of 25 in two-dimensional and 66 in three-dimensional was obtained.