Structure-preserving local discontinuous Galerkin discretization of conformational conversion systems

TL;DR

This paper introduces a novel structure-preserving numerical scheme for a two-state conformational conversion system, ensuring positivity, boundedness, and entropy stability, with proven convergence and validated numerical results.

Contribution

The paper develops a new structure-preserving discretization combining local discontinuous Galerkin and backward Euler methods, with theoretical proofs of stability and convergence.

Findings

The scheme enforces positivity and boundedness at the discrete level.

A discrete entropy-stability inequality is established.

Numerical results confirm theoretical properties and practical effectiveness.

Abstract

We investigate a two-state conformational conversion system and introduce a novel structure-preserving numerical scheme that couples a local discontinuous Galerkin space discretization with the backward Euler time-integration method. The model is first reformulated in terms of auxiliary variables involving suitable nonlinear transformations, which allow us to enforce positivity and boundedness at the numerical level. Then, we prove a discrete entropy-stability inequality, which we use to show the existence of discrete solutions, as well as to establish the convergence of the scheme by means of some discrete compactness arguments. As a by-product of the theoretical analysis, we also prove the existence of global weak solutions satisfying the system's physical bounds. Numerical results validate the theoretical results and assess the capabilities of the proposed method in practice.