Problems of nonlinear computational instability in evolution equations

Some problems of nonlinear computational instability are discussed in this article, which are shown as follows: 1) Three types of representative evolution equations are analyzed, and the close relationship between the nonlinear computational stability or instability in their corresponding difference equations and the properties of their solution is revealed. 2) The problem of nonlinear computational instability in conservative differencing equations with the periodic boundary condition is further discussed, and some effective ways to avoid nonlinear computational instability are proposed. 3) The problem of nonlinear computational instability in non-conservative difference equations with the aperiodic boundary condition is focused on by using nonlinear advection equations as examples, and “synthetic analysis method” is given to judge their computational stability.