Volume 5, Issue 1
Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

Eveline Rosseel, Nico Scheerlinck & Stefan Vandewalle

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 62-84.

Published online: 2012-05

[An open-access article; the PDF is free to any online user.]

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  • Abstract

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

  • Keywords

Multigrid, stochastic Galerkin finite element method, reaction-diffusion problems, implicit Runge-Kutta method and PDEs with random coefficients.

  • AMS Subject Headings

35K57, 35Q92, 65M55, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-62, author = {}, title = {Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {1}, pages = {62--84}, abstract = {

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m12si04}, url = {http://global-sci.org/intro/article_detail/nmtma/5928.html} }
TY - JOUR T1 - Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 62 EP - 84 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2011.m12si04 UR - https://global-sci.org/intro/article_detail/nmtma/5928.html KW - Multigrid, stochastic Galerkin finite element method, reaction-diffusion problems, implicit Runge-Kutta method and PDEs with random coefficients. AB -

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

Eveline Rosseel, Nico Scheerlinck & Stefan Vandewalle. (2020). Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients. Numerical Mathematics: Theory, Methods and Applications. 5 (1). 62-84. doi:10.4208/nmtma.2011.m12si04
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