Nonlinear difference schemes for linear partial differential equations

We discuss a nonlinear difference scheme for approximating the solution of the initial value problem for linear partial differential equations. At each time step of the calculation the method proceeds by processing the data and determining the best possible scheme to use for that step, according to an optimization criterion to be described. We show that the method is stable and convergent applicating it on the heat equation. In all cases considered the nonlinear method was more accurate than the classical methods. © 1973 Springer-Verlag.