Structure-preserving local discontinuous Galerkin discretization of conformational conversion systems

TL;DR

The paper develops a structure-preserving numerical framework for a two-state conformational conversion system by reformulating the model in entropy variables and discretizing with a backward Euler-LDG scheme. It proves a discrete entropy-stability inequality, establishes the existence of discrete solutions, and demonstrates convergence to a weak solution of the continuous problem as the mesh is refined and the penalty/time-step parameters vanish. The method enforces positivity and bounds at the discrete level, while maintaining a parallelizable flux formulation with no nonlinearities on interfaces. Numerical experiments validate the theoretical results, showing optimal convergence rates, accurate traveling-wave behavior, and correct qualitative dynamics under varied diffusion scenarios. This work thus provides a rigorous, structure-preserving tool for simulating spatially distributed conformational dynamics with reliable long-time behavior and physical consistency.

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.

Figures (7)

• Figure 1: Test case 1: computed errors and convergence rates w.r.t. the mesh size $h$.
• Figure 2: Test case 1: computed errors and convergence rates w.r.t. the polynomial degree $\ell$.
• Figure 3: Test case 2: Initial conditions ($t=0$) and solutions at $t=1$ (first row) for different polynomial degrees $\ell=1,...,5$ with associated approximation errors (second row) for the variables $p$ (a) and $q$ (b).
• Figure 4: Test case 1: computed errors and convergence rates w.r.t. the polynomial degree $\ell$.
• Figure 5: Test case 3: numerical solutions $q_h^{(n)}$ (first column of each panel) and $p_h^{(n)}$ (second column of each panel) at different times in the case of stable focus (left panel) and stable node equilibrium (right panel).

Theorems & Definitions (27)

• Remark 2.1: Initial datum $p_0$
• Remark 2.2: The Fisher-Kolmogorov equation
• Remark 2.3: The heterodimer model
• Remark 2.4: Possible degeneracy
• Remark 2.5: Auxiliary variables
• Remark 2.6: Space of $d$-vector-valued polynomials
• Remark 2.7: Role of the penalty term
• Remark 2.8: Equation \ref{['EQ::VARIATIONAL_SIGMA']} uniquely determines $\boldsymbol{\sigma}_{\star,h}^{(n + 1)}$
• Remark 2.9: Formulation in terms of the $\mathbf{W}$ unknowns only
• Proposition 3.1: A bound for the reaction term