Observations on an adaptive moving grid method for one-dimensional systems of partial differential equations

Recently a scheme has been proposed for choosing a moving mesh based on minimizing the time rate of change of the solution in the moving coordinates for one-dimensional systems of PDEs. In this paper we show how to apply this idea to systems where the time derivatives cannot be solved for explicitly, writing the moving mesh equations in an implicit form. We give a geometrical interpretation of the scheme which exposes some of its weaknesses, and suggest some modifications based on this interpretation which increase the efficiency of the scheme. Finally, we present some numerical experiments which illustrate how well the resulting method works.