Efficient bound preserving and asymptotic preserving semi-implicit schemes for the fast reaction-diffusion system

TL;DR

Addressing stiff fast reaction-diffusion systems with $u_\\epsilon$, $v_\\epsilon$, $p_\\epsilon$ where the reaction rate scales as $O(1/\\epsilon)$ and diffusion is $O(1)$, the paper analyzes the asymptotic Stefan-limit behavior as $\\epsilon \to 0$. It introduces a first-order semi-implicit time discretization that preserves positivity and bounds and a second-order semi-implicit Runge–Kutta scheme, both designed to capture interface propagation without resolving the small parameter. The authors establish discrete positivity, bound preservation, $L^2$ stability, and linearized stability near the limit, along with residual analyses indicating asymptotic preservation to the Stefan problem. Numerical tests in 1D and 2D validate convergence, bound-preserving properties, and sharp-interface capture across varying $\\epsilon$, demonstrating the practical utility for simulating melting and solidification processes in multi-scale heat-transfer problems.

Abstract

We consider a special type of fast reaction-diffusion systems in which the coefficients of the reaction terms of the two substances are much larger than those of the diffusion terms while the diffusive motion to the substrate is negligible. Specifically speaking, the rate constants of the reaction terms are $O(1/ε)$ while the diffusion coefficients are $O(1)$ where the parameter $ε$ is small. When the rate constants of the reaction terms become highly large, i.e. $ε$ tends to 0, the singular limit behavior of such a fast reaction-diffusion system is inscribed by the Stefan problem with latent heat, which brings great challenges in numerical simulations. In this paper, we adopt a semi-implicit scheme, which is first-order accurate in time and can accurately approximate the interface propagation even when the reaction becomes extremely fast, that is to say, the parameter $ε$ is sufficiently small. The scheme satisfies the positivity, bound preserving properties and has $L^2$ stability and the linearized stability results of the system. For better performance on numerical simulations, we then construct a semi-implicit Runge-Kutta scheme which is second-order accurate in time. Numerous numerical tests are carried out to demonstrate the properties, such as the order of accuracy, positivity and bound preserving, the capturing of the sharp interface with various $ε$ and to simulate the dynamics of the substances and the substrate, and to explore the heat transfer process, such as solid melting or liquid solidification in two dimensions.

Figures (8)

• Figure 1: Left: First-order semi-implicit scheme: accuracy order check in $\tau$ with $\epsilon = 1$ and $h = 1.25 \times 10^{-2}, \tau = 2^{-j} \tau_0, j = 0, 1, \cdots, 9, 10$. Right: First-order semi-implicit scheme: accuracy order check in $h$ with $\epsilon = 1$ and $\tau = 6.25 \times 10^{-3}, h = 2^{-j} h_0, j = 0, 1, \cdots, 5, 6$.
• Figure 2: Left: First-order semi-implicit scheme: accuracy order check in $\tau$ with $\epsilon = 10^{-2}$ and $h = 1.25 \times 10^{-2}, \tau = 2^{-j} \tau_0, j = 0, 1, \cdots, 9, 10$. Right: Second-order semi-implicit Runge--Kutta scheme: accuracy order check in $\tau$ with $\epsilon = 10^{-2}$ and $h = 6.25 \times 10^{-3}, \tau = 2^{-j} \tau_0, j = 0, 1, \cdots, 9, 10$.
• Figure 3: First-order semi-implicit scheme with $\epsilon = 10^{-2}$ and $h = 0.0125, \tau = 5 \times 10^{-6}$.
• Figure 4: First-order semi-implicit scheme: solutions $u, v, p, w$ with $\epsilon = 10^{-4}$ and $h = 6.25 \times 10^{-3}$, $\tau = 5 \times 10^{-6}$.
• Figure 5: First-order semi-implicit scheme: the enthalpy function $w$ at $t = 1$ with various value of $\epsilon$ with $\tau = 5 \times 10^{-6}$ and $h = 0.0125$.

Theorems & Definitions (10)

• Lemma 2.1: Bound preserving
• proof
• Lemma 2.2: $L^2$ stability
• proof
• Lemma 2.3: Stability near the limit behavior for the linearized setting
• proof
• Theorem 2.1: Bound preserving
• proof
• Theorem 2.2: $L^2$ stability
• proof