Journal
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume 381, Issue -, Pages -Publisher
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2021.113746
Keywords
Supplementary variable method; Thermodynamically consistent models; Gradient flows; Energy-production-rate preserving schemes; Finite difference methods; Pseudo-spectral methods
Funding
- Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, China [202001, 202002]
- China Postdoctoral Science Foundation [2020M670116]
- Natural Science Foundation of Jiangsu Province, China [BK20180413]
- National Natural Science Foundation of China [11801269, 12071216, 11971051, NSAF-U1930402]
- National Science Foundation of US [DMS-1815921, OIA-1655740]
- DOE [DE-SC0020272]
- GEAR award from SC EPSCoR/IDeA Program
- U.S. Department of Energy (DOE) [DE-SC0020272] Funding Source: U.S. Department of Energy (DOE)
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A paradigm called the supplementary variable method (SVM) is presented to develop thermodynamically consistent numerical algorithms for thermodynamically consistent partial differential equation systems. The study shows that by adding appropriate supplementary variables, the extended system can maintain consistency and solvability, with the new schemes performing better overall compared to existing schemes.
We present a paradigm for developing thermodynamical consistent numerical algorithms for thermodynamically consistent partial differential equation (TCPDE) systems, called the supplementary variable method (SVM). We add a proper number of supplementary variables to the TCPDE system coupled with its energy equation and other deduced equations through perturbations to arrive at a consistent, well-determined, solvable and structurally stable system. The extended system not only reduces to the TCPDE system at specific values of the supplementary variables, but also allows one to retain consistency and solvability after a consistent numerical approximation. Among virtually infinite many possibilities to add the supplementary variables, we present two that maintain thermodynamical consistency in the extended system before and after the approximation. A pseudo-spectral method is used in space to arrive at fully discrete schemes. The new schemes are compared with the energy stable SAV scheme and the fully implicit Crank-Nicolson scheme. The numerical results favor the new schemes in the overall performance. (C) 2021 Elsevier B.V. All rights reserved.
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