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F.W. Yang¹, C.E. Goodyer², M.E. Hubbard³ and P.K. Jimack⁴

¹Department of Mathematics, University of Sussex, Brighton, United Kingdom
²NAG Ltd., Wilkinson House, Jordan Hill Road, Oxford, United Kingdom
³School of Mathematical Sciences, University of Nottingham, United Kingdom
⁴School of Computing, University of Leeds, United Kingdom

doi:10.4203/ccp.107.39

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F.W. Yang, C.E. Goodyer, M.E. Hubbard, P.K. Jimack, "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", in , (Editors), "Proceedings of the Fourth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 39, 2015. doi:10.4203/ccp.107.39

Keywords: parallel, adaptive mesh refinement, finite differences, implicit, multigrid.

Summary

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 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 differentiation 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 paper illustrates the application of this new software framework to a challenging application, namely a multi-phase-field model of tumour growth. We show some typical simulations for tumour growth, 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).

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 107 PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED, GRID AND CLOUD COMPUTING FOR ENGINEERING Edited by: Paper 39 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 F.W. Yang¹, C.E. Goodyer², M.E. Hubbard³ and P.K. Jimack⁴ ¹Department of Mathematics, University of Sussex, Brighton, United Kingdom ²NAG Ltd., Wilkinson House, Jordan Hill Road, Oxford, United Kingdom ³School of Mathematical Sciences, University of Nottingham, United Kingdom ⁴School of Computing, University of Leeds, United Kingdom doi:10.4203/ccp.107.39 purchase the full-text of this paper Full Bibliographic Reference for this paper F.W. Yang, C.E. Goodyer, M.E. Hubbard, P.K. Jimack, "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", in , (Editors), "Proceedings of the Fourth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 39, 2015. doi:10.4203/ccp.107.39 Keywords: parallel, adaptive mesh refinement, finite differences, implicit, multigrid. Summary 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 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 differentiation 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 paper illustrates the application of this new software framework to a challenging application, namely a multi-phase-field model of tumour growth. We show some typical simulations for tumour growth, 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). purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £45 +P&P)
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