Surprising computations

摘要

In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We examine several instances of such computations. These include (i) solving Hamiltonian ODE systems using almost conservative explicit Runge-Kutta methods, (ii) applying splitting methods for the nonlinear Schrodinger equation, and (iii) applying strong stability preserving Runge-Kutta methods in conjunction with weighted essentially non-oscillatory semi-discretizations for nonlinear conservation laws with discontinuous solutions. For each problem and method class we present a simple numerical example that yields results that in our experience many active researchers are finding unexpected and unintuitive. Each numerical example is then followed by an explanation and a resolution of the practical problem.