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

Yang, F W, Goodyer, C E, Hubbard, M E and Jimack, P K (2017) An optimally efficient technique for the solution of systems of nonlinear parabolic partial differential equations. Advances in Engineering Software, 103. pp. 65-84. ISSN 0965-9978

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Abstract

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.

Item Type: Article
Keywords: Parallel, Adaptive mesh refinement, Finite difference, Implicit, Multigrid, Thin film flow, Tumour growth
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA0101 Elementary mathematics. Arithmetic
Q Science > QA Mathematics > QA0076 Computer software
Depositing User: Feng Wei Yang
Date Deposited: 08 Jun 2016 14:12
Last Modified: 15 Mar 2017 09:41
URI: http://sro.sussex.ac.uk/id/eprint/61396

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Unravelling new mathematics for 3D cell migrationG1438LEVERHULME TRUSTRPG-2014-149