Structure-preserving Local Discontinuous Galerkin method for nonlinear cross-diffusion systems

TL;DR

This work develops a structure-preserving Local Discontinuous Galerkin method for nonlinear cross-diffusion systems by reformulating the problem in entropy variables and introducing auxiliary fields. The method preserves the entropy structure and enforces positivity/boundedness of the physical variables through a nonlinear entropy-variable transformation, while keeping nonlinearities off differential operators to enhance efficiency. A fully discrete, regularized scheme with backward Euler time stepping is analyzed for existence, entropy stability, and convergence to weak solutions of the continuous problem, with rigorous results under a set of DG-norm and regularization assumptions. Numerical experiments in one and two dimensions demonstrate high-order convergence, entropy stability, and the ability to capture complex phenomena such as Turing patterns, validating the approach for practical cross-diffusion systems.

Abstract

We present and analyze a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time-stepping scheme. The proposed method makes use of the underlying entropy structure of the system, expressing the main unknown in terms of the entropy variable by means of a nonlinear transformation. Such a transformation allows for imposing the physical positivity or boundedness constraints on the approximate solution in a strong sense. A key advantage of our scheme is that nonlinearities do not appear explicitly within differential operators or interface terms in the scheme, which significantly improves its efficiency and eases its implementation. We prove the existence of discrete solutions and their asymptotic convergence to a weak solution to the continuous problem. Numerical results for some one- and two-dimensional problems illustrate the accuracy and entropy stability of the proposed method.

Figures (10)

• Figure 1: Example of a two dimensional domain $\Omega$ (in yellow). Left panel: Triangular mesh $\mathcal{T}_h$ of $\Omega$ and an interior point $(\overline{x}, \overline{y})$ (depicted with a red dot) of some element $K\in \mathcal{T}_h$. Right panel: Auxiliary domain $\widetilde{\Omega}$ (in blue) and auxiliary mesh $\widetilde{\mathcal{T}}_h$.
• Figure 2: Example of vertices of $\widetilde{\mathcal{T}}_h$ lying along $\partial \widetilde{\Omega}^{\overline{x}}$. The red dot has the coordinates $(\overline{x}, \overline{y})$; the green dots have the coordinates $\{(\overline{x}, y_j)\}_{j = 1}^{\ell}$ for some $\ell \in \mathbb{N}$; the purple dot belongs to $\partial \Omega$ and has the coordinates $(\overline{x}, y^{\partial})$.
• Figure 3: Illustration of the two types of points in the set $\{(\overline{x}, y_j)\}_{j = 1}^{\ell}$ used in the bound on $J_4$.
• Figure 4: Example of the auxiliary segments $\widetilde{\Gamma}$ (left panel) and $\Gamma$ (right panel) in the bound on $J_5$. The yellow dot has the coordinates $(\hat{x}, \hat{y})$ and is a vertex of $\Omega$. The cyan dot has the coordinates $(\hat{x}_1, \hat{x}_2)$
• Figure 5: $h$-convergence of the errors in \ref{['EQN::ERRORS-1D']} at the final time $T = 1$ for the porous medium equation with exact solution $\rho$ in \ref{['EQN::EXACT-SOL-POROUS']}. The numbers in the yellow rectangles denote the experimental rates of convergence.

Theorems & Definitions (31)

• Remark 2.1: Choice of $\alpha_F$
• Remark 2.2: Computation of $\underaccent{\bar{}}{\boldsymbol{\sigma}}_h$
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Remark 2.6
• Remark 2.7: Constant diffusion tensor $A$