Modified equation techniques for reactive-diffusive systems. I: Explicit, implicit and quasilinear methods (Q1087001)

A family of modified equation techniques for one-dimensional reaction- diffusion equations is developed and applied to study of a single reaction-diffusion equation which has an exact travelling wave solution. Modified equation methods are also applied to the study the propagation of a laminar flame through an ozone-oxygen mixture. It is shown that modified implicit techniques are less accurate than the standard implicit method if the time derivatives are evaluated by means of a first-order accurate formula. If the time derivatives are approximated by second- order accurate expressions, modified equation methods yield \(L^ 2\) norm errors which are almost independent of the time step used in the calculations; they are also independent of time. Large accuracy improvements can be obtained by using modified equation methods. These improvements are achieved at the expense of involving five grid points and requiring iterations if quasilinear techniques are used in the calculations. A linear stability analysis of the modified equation methods presented in this paper shows that the modified explicit scheme requires smaller time steps than the standard explicit method.