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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.

  • AMS Subject Headings

65M12,35K59,74N05

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-661, author = {R. H. W. Hoppe and J. J. Winkle}, 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 AU - R. H. W. Hoppe & J. J. Winkle 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 and 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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