Volume 12, Issue 3
A Splitting Scheme for the Numerical Solution of the KWC System

R. H. W. Hoppe & J. J. Winkle

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 661-680.

Published online: 2019-04

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  • Abstract

We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.

  • Keywords

Crystallization, Kobayashi-Warren-Carter system, splitting method

  • AMS Subject Headings

65M12,35K59,74N05

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-661, author = {}, title = {A Splitting Scheme for the Numerical Solution of the KWC System}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {3}, pages = {661--680}, abstract = {

We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0050}, url = {http://global-sci.org/intro/article_detail/nmtma/13125.html} }
TY - JOUR T1 - A Splitting Scheme for the Numerical Solution of the KWC System JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 661 EP - 680 PY - 2019 DA - 2019/04 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2018-0050 UR - https://global-sci.org/intro/article_detail/nmtma/13125.html KW - Crystallization, Kobayashi-Warren-Carter system, splitting method AB -

We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.

R. H. W. Hoppe & J. J. Winkle. (2019). A Splitting Scheme for the Numerical Solution of the KWC System. Numerical Mathematics: Theory, Methods and Applications. 12 (3). 661-680. doi:10.4208/nmtma.OA-2018-0050
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