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In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented.
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On a positive-preserving, energy-stable numerical scheme to mass-action kinetics with detailed balance
In this paper, we provide a detailed theoretical analysis of the numerical scheme introduced in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436:110253, 2021] for the reaction kinetics of a class of chemical reaction networks that satisfies detailed balance condition. In contrast to conventional numerical approximations, which are typically constructed based on ordinary differential equations (ODEs) for the concentrations of all involved species, the scheme is developed using the equations of reaction trajectories, which can be viewed as a generalized gradient flow of a physically relevant free energy. The unique solvability, positivity-preserving, and energy-stable properties are proved for the general case involving multiple reactions, under a mild condition on the stoichiometric matrix.
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- Award ID(s):
- 2410740
- PAR ID:
- 10625705
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Communications in Mathematical Sciences
- Volume:
- 23
- Issue:
- 5
- ISSN:
- 1539-6746
- Page Range / eLocation ID:
- 1447 to 1458
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/CMS.250517004551
