An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems
For researchers solving cross-diffusion systems with gradient flow structure, this provides an entropy-stable and positivity-preserving numerical scheme, though it is an incremental extension of prior work.
This work extends a previous discontinuous Galerkin method to cross-diffusion gradient flow systems, achieving entropy stability and positivity preservation. Numerical examples demonstrate the method's performance on 1D and 2D Cartesian meshes.
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.