An optimally efficient technique for the solution of systems of nonlinear parabolic partial differential equations

- This paper describes a new software tool for the efficient solution of systems of linear and nonlinear partial differential equations of parabolic type. Specifically, this software is designed to provide optimal computational performance for multiscale problems, which require hightly stable, implicit, time-stepping schemes combined with a parallel implementation of adaptivity in both space and time.
- An optimally efficient nonlinear multigrid solver is implemented to obtain computational solutions to a very high resolution with relatively modest computational resources.
- A thin film flow model of a spreading viscous droplet and a multi-phase-field model of tumour growth are included to demonstrate the flexibility and robustness of our software tool.
- We also discuss 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.

This paper describes a new software tool that has been developed for the efficient solution of systems of linear and nonlinear partial differential equations (PDEs) of parabolic type. Specifically, the software is designed to provide optimal computational performance for multiscale problems, which require highly stable, implicit, time-stepping schemes combined with a parallel implementation of adaptivity in both space and time. By combining these implicit, adaptive discretizations with an optimally efficient nonlinear multigrid solver it is possible to obtain computational solutions to a very high resolution with relatively modest computational resources. The first half of the paper describes the numerical methods that lie behind the software, along with details of their implementation, whilst the second half of the paper illustrates the flexibility and robustness of the tool by applying it to two very different example problems. These represent models of a thin film flow of a spreading viscous droplet and a multi-phase-field model of tumour growth. 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.