Towards a Three-Dimensional Parallel, Adaptive, Multilevel Solver for the Solution of Nonlinear, Time-Dependent, Phase-Change Problems

One of the fundamental open challenges that computational engineering practitioners continue to face is that of combining modern numerical algorithms, such as mesh adaptivity and efficient multilevel solvers, with a parallel implementation that permits scalable performance on large numbers of cores. The fundamental problems that must be overcome include the need to maintain an effective load balance that responds dynamically to local mesh refinement and/or coarsening, and the difficulty of obtaining good parallel efficiencies for those operations that take place at the coarsest levels of a multilevel solver. In this paper we report on our ongoing research which seeks to make progress towards resolving some of these challenges through the application of a parallel, adaptive, multilevel approach to the solution of a highly challenging class of phase-change problems involving a very wide range of length and time scales. The mathematical models that we seek to solve take the form of systems of highly nonlinear, stiff, parabolic partial differential equations (PDEs) in three space dimensions. The solutions of these PDEs require extremely high spatial resolution near to a moving interface and so mesh refinement is essential. Furthermore, the stiffness of the systems implies that implicit time-stepping must be used and the resulting nonlinear algebraic systems are solved using a nonlinear multigrid approach on these locally refined meshes. In the first part of the paper we provide details of the phase-change problems that we study, including their mathematical formulation, and we illustrate the need for both adaptive mesh refinement and fully implicit time-stepping, even for problems in just two space dimensions. The desire to move to three space dimensions provides the primary motivation for extending our techniques to parallel architectures, which forms the substance of the second half of the paper. We introduce an open source software tool, PARAMESH [1], and describe how we have developed it to allow its application to our problems, involving the use of nonlinear multigrid on adaptively refined meshed in three dimensions, in parallel. Initially we use some simple linear model equations in order to begin to assess the performance of our software, and this is followed by some results computed for nonlinear PDE systems which model a typical rapid solidification problem. The paper concludes with a summary and a short description of our planned future work.