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Volume 33, Issue 2
A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

Weiwen Wang & Chuanju Xu

Commun. Comput. Phys., 33 (2023), pp. 477-510.

Published online: 2023-03

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  • Abstract

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

  • AMS Subject Headings

80A22, 35Q79, 65L06, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-477, author = {Wang , Weiwen and Xu , Chuanju}, title = {A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {2}, pages = {477--510}, abstract = {

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0183}, url = {http://global-sci.org/intro/article_detail/cicp/21497.html} }
TY - JOUR T1 - A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems AU - Wang , Weiwen AU - Xu , Chuanju JO - Communications in Computational Physics VL - 2 SP - 477 EP - 510 PY - 2023 DA - 2023/03 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0183 UR - https://global-sci.org/intro/article_detail/cicp/21497.html KW - Thermal phase change problem, gradient flows, unconditional energy stability, auxiliary variable, Runge-Kutta methods, phase field. AB -

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

Wang , Weiwen and Xu , Chuanju. (2023). A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems. Communications in Computational Physics. 33 (2). 477-510. doi:10.4208/cicp.OA-2022-0183
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