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

TL;DR

This paper introduces a structure-preserving Local Discontinuous Galerkin method for nonlinear cross-diffusion systems that ensures physical constraints and improves efficiency by leveraging entropy structures and nonlinear transformations.

Contribution

The paper develops a novel entropy-based LDG scheme that maintains positivity and boundedness, with proven convergence and demonstrated numerical accuracy for complex systems.

Findings

Ensures physical positivity constraints in solutions.

Proves convergence to weak solutions.

Demonstrates accuracy and entropy stability numerically.

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.