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

作　　者：GONG Jing WANG Bin JI Zhong-Zhen 

机构地区：[1]State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences [2]Mathematics and Physics Department, North China Electric Power University

出　　处：《Atmospheric and Oceanic Science Letters》2013年第6期423-427,共5页大气和海洋科学快报（英文版）

基　　金：the Ministry of Science and Technology of China for the National Basic Research Program of China(973 Program,Grant No.2011CB309704)

摘　　要：In 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 solution, while the RTI scheme outperforms the LF scheme and CSCD scheme only in the first 1000 steps and then becomes the worst in terms of RMSE up to the 2500th step. It is concluded that reasonable consideration of accuracy, square conservation, and historical observations is also critical for good performance of a finite difference scheme for solving nonlinear equations.In 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 solution, while the RTI scheme outperforms the LF scheme and CSCD scheme only in the first 1000 steps and then becomes the worst in terms of RMSE up to the 2500th step. It is concluded that reasonable consideration of accuracy, square conservation, and historical observations is also critical for good performance of a finite difference scheme for solving nonlinear equations.

关 键 词：three-step difference scheme NONLINEAR square conservation accuracy historical observations 

分 类 号：P413[天文地球—大气科学及气象学]