An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems
An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems
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DOI：
10.3934/krm.2019033
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发表时间：
2018-10
影响因子：
1
通讯作者：
Zheng Sun;J. Carrillo;Chi-Wang Shu
中科院分区：
文献类型：
--
作者：
Zheng Sun;J. Carrillo;Chi-Wang Shu
作为文献[ 41 ]中工作的推广，我们发展了一个求解具有形式梯度流结构的交叉扩散方程组的间断Galerkin方法。这些系统与非增熵泛函相关联。对于一类问题，解的正性（非负性）也是人们所期望的，这是物理模型所隐含的，对熵结构至关重要。我们提出的半离散数值格式是熵稳定的。此外，该格式在许多情况下也与[ 43 ]中的保正过程兼容，因此所得到的全离散格式能够产生非负解。该方法既适用于一维问题，也适用于笛卡尔网格上的二维问题。数值例子来检验该方法的性能。
As an extension of our previous work in [ 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 [ 43 ] 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.