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arXiv · 1810.03221

An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems

Zheng Sun·José Antonio Carrillo·Chi-Wang Shu
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Abstract

As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in Zhang (2017) [42] in many scenarios. Hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

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Zheng Sun, José Antonio Carrillo, Chi-Wang Shu. 2018-10-07. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. https://arxiv.org/abs/1810.03221

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