Three-Step Difference Scheme for Solving Nonlinear Time-Evolution Partial Differential Equations

Abstract

AbstractIn this paper, a special three-step difference scheme is applied to the solution of nonlinear time-evolution equations, whose coefficients are determined according to accuracy constraints, necessary conditions of square conservation, and historical observation information under the linear supposition. As in the linear case, the schemes also have obvious superiority in overall performance in the nonlinear case compared with traditional finite difference schemes, e.g., the leapfrog (LF) scheme and the complete square conservation difference (CSCD) scheme that do not use historical observations in determining their coefficients, and the retrospective time integration (RTI) scheme that does not consider compatibility and square conservation. Ideal numerical experiments using the one-dimensional nonlinear advection equation with an exact solution show that this three-step scheme minimizes its root mean square error (RMSE) during the first 2500 integration steps when no shock waves occur in the exact so...