Multigrid solution methods for nonlinear time-dependent systems

An efficient, accurate and reliable numerical solver is essential for solving complex mathematical models and obtaining their computational approximations. The solver presented in this work is built upon nonlinear multigrid with the full approximation scheme (FAS). Its implementation is achieved, in part, using a complex, open source software library PARAMESH, and the resulting numerical solver, Campfire, also combines with adaptive mesh refinement, adaptive time stepping and parallelization through domain decomposition.
There are five mathematical models considered in this work, ranging from applications such as binary alloy solidification and fluid dynamics to a multi-phase-field model of tumour growth. These mathematical models consist of nonlinear, time-dependent and coupled partial differential equations (PDEs). Using our adaptive, parallel multigrid solver, together with finite difference method (FDM) and backward differentiation formulas (BDF), we are able to solve all five models in computationally demanding 2-D and/or 3-D situations.
Due to the choice of second order central finite difference and second order BDF2 method, we obtain, and demonstrate, solutions with an overall second order convergence rate and optimal multigrid convergence. In the case of the multi-phase-field model of tumour growth, this has not previously been achieved. The novelties of our work also include solving the model of binary alloy solidification with a time-dependent temperature field in 3-D for the first time; implementing non-time-dependent equations alongside the coupled time-dependent partial differential equations (and increasing the range of boundary conditions) to significantly increase the generality and range of applicability of the described multigrid solver; improving the efficiency of the implementation of the solver through multiple developments; and introducing penalty terms to smoothly control the behaviour of phase variables where their range of valid values is constrained.