A Splitting Moving Mesh Method for 3-D Quenching and Blow-Up Problems

We present a splitting moving mesh method for multi-dimensional reaction-diffusion problems with nonlinear forcing terms over rectangular domains. The structure of the adaptive algorithm is an elegant combination of an operator splitting and one-dimensional moving mesh. It is motivated by the nature of splitting method, which splits a multi-dimensional problems into a few one-dimensional problems. Therefore, the temporal and the spatial adaptations are adopted based on one-dimensional arc-length equidistributed rule. The method not only keeps the advantage of splitting methods on reducing computational cost but also simplifies the implementation of moving mesh techniques. The method is first presented in [20] for the 2-D quenching problems using Peaceman-Rachford splitting. In this paper, we apply the splitting moving mesh method to both the 3-D blow-up and the 3-D quenching singularity using Douglas splitting. 3-D numerical examples will be given to demonstrate the good performance of the method.