Computational Stability and Time Truncation of Coupled Nonlinear Equations with Exact Solutions

A general numerical integration formula is presented that generates many of the commonly used one-dimensional finite-difference schemes. A number of these schemes are tested on a simple wave equation; three implicit and three explicit are chosen for further analysis with a nonlinear set of equations with known solutions. A seventh method of the implicit type not requiring iteration is also tested. A transformation is developed that allows the removal of linear terms from the nonlinear equations, thereby avoiding truncation of the linear terms. The results of the analysis show that energy components may have large errors when the total energy shows essentially none, and phase errors may be quite serious without indication from linear analysis. By treating the uncoupled linear terms exactly (no truncation), significant improvement in the numerical solutions ensues. The multilevel implicit schemes give superior results and are to be recommended if computing time is not a criterion. Great care must be taken in interpreting the linear stability criterion. The critical truncation increment should be considerably reduced to avbid significant truncation errors, especially for long time integrations.