A semi-adaptive finite difference method for simulating two-sided fractional convection-diffusion quenching problems

TL;DR

This work studies quenching in a one-dimensional, two-sided fractional convection-diffusion equation with a nonlinear source $p(v)=(\kappa-v)^{- heta}$ using a semi-adaptive finite difference method based on weighted Grünwald approximations for the fractional derivatives. A semi-discretized system $\frac{dv}{dt}=S v+p(v)$ is advanced to a fully discrete scheme with adaptive time stepping, analyzed for positivity, monotonicity, and stability under specific conditions on the fractional order $\sigma$ and the time–space mesh ratio. The authors systematically explore how the fractional order, convection coefficient, and nonlinearity influence quenching time $T_a$, quenching location $x^*$, and critical length $a^*$ through four simulation experiments, and they validate convergence via Milne-based devices. The results highlight nonlocal effects on quenching, provide benchmarks against integer-order and existing estimates, and lay groundwork for high-dimensional extensions and physics-informed numerical strategies for fractional PDE quenching problems.

Abstract

This paper investigates quenching solutions of an one-dimensional, two-sided Riemann-Liouville fractional order convection-diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in conjunction with standard and shifted Grünwald formulas. The advective term is handled utilizing a straightforward Euler formula, resulting in a semi-discretized system of nonlinear ordinary differential equations. The conservativeness of the proposed scheme is rigorously proved and validated through simulation experiments. The study is further advanced to a fully discretized, semi-adaptive finite difference method. Detailed analysis is implemented for the monotonicity, positivity and stability of the scheme. Investigations are carried out to assess the potential impacts of the fractional order on quenching location, quenching time, and critical length. The computational results are thoroughly discussed and analyzed, providing a more comprehensive understanding of the quenching phenomena modeled through two-sided fractional order convection-diffusion problems.

Figures (11)

• Figure 1: LEFT: Relationship between $a^{*}$ and $b$ ($\sigma=2$); RIGHT: Relationship between relative error $\left|a^{*}-a^{*S}\right|/\left|a^{*S}\right|$ and parameter $b.$
• Figure 2: LEFT: Relationship between $a^{*}$ and $b$ ($\theta=1$); RIGHT: Relationship between $a^{*}$ and $\sigma$ ($\theta=1$).
• Figure 3: LEFT: Relative error of computed quenching time $T_{a}$ from $T_{a}^{S},T_{a}^{M},T_{a}^{L}$ ($a=\pi,\sigma=2,\theta=1$); RIGHT: Relative error of computed quenching time $T_{a}$ from $T_{a}^{S},T_{a}^{M},T_{a}^{L}$ ($a=2,\sigma=2,\theta=1$).
• Figure 4: A monotonic convergence of $T_{a}$ as $a\rightarrow\infty.$
• Figure 5: LEFT: Dependence of $T_{a}$ on $b.$ RIGHT: Dependence of $T_{a}$ on the fractional derivative order $\sigma.$

Theorems & Definitions (9)

• Definition 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Definition 3.2
• Theorem 3.1
• Lemma 3.4
• Theorem 3.2
• Corollary 3.1