Parallel Implementation of an Adaptive, Multigrid Solver for the Implicit Solution of Nonlinear Parabolic Systems, with Application to a Multi-Phase-Field Model for Tumour Growth

The first half of the paper provides an overview of a new engineering software tool that is designed for the efficient solution of problems that may be modeled as systems of linear and nonlinear partial differential equations (PDEs) of parabolic type. Our tool is built upon the PARAMESH library, [15], which provides hierarchical mesh adaptivity in parallel in two and three dimensions. Our discretizations are based upon cell-centred finite difference schemes in space and implicit multi-step methods in time (primarily the second order backward differential formula (BDF2)). This results in the need to solve a nonlinear algebraic system at each time step, and we have implemented an optimal nonlinear multigrid method based upon full approximation scheme (FAS). The second half of the presentation illustrates the application of this new software framework to a challenging application, namely a multi-phase-field model of tumour growth [18]. We show some typical simulations for growth of the model tumours, and these results demonstrate second-order convergence in both space and time. We conclude with a discussion of the challenges of obtaining highly scalable parallel performance for a software tool that combines both local mesh adaptivity (requiring efficient dynamic load-balancing) and a multigrid solver (requiring careful implementation of coarse grid operations and inter-grid transfer operations in parallel).