Efficient time integrators in the numerical method of lines

The numerical method of lines has long been acknowledged as a very powerful approach to the numerical solution of time dependent partial differential equations. This method, in its original form, involved making a simple approximation to the space derivatives, and by so doing reducing the problem to that of solving a system of initial value ordinary differential equations, and then using a “black box” package as the time integrator. However in the past twenty years or so, moving mesh algorithms in space have been developed and this allows much more challenging problems (for example those with moving fronts) to be solved efficiently and reliably. Regridding of the space variables poses special problems for the time integrator since sufficient back information to allow multistep formulae to run at high order is not available immediately after the regridding has been performed . In this paper we survey some of the options available for the time integration when using a moving grid method of lines code. In particular we derive 'Runge-Kutta starters' for use after grid adaptation has been carried out and we show how a considerable saving in computational effort can be made if just a few spatial points are moved during each regridding.