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

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posted on 2023-06-09, 01:36 authored by F W Yang, C E Goodyer, M E Hubbard, P K Jimack
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.

Funding

Unravelling new mathematics for 3D cell migration; G1438; LEVERHULME TRUST; RPG-2014-149

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Advances in Engineering Software

ISSN

0965-9978

Publisher

Elsevier

Volume

103

Page range

65-84

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2016-06-08

First Open Access (FOA) Date

2017-01-14

First Compliant Deposit (FCD) Date

2016-06-08

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