LDG method for solving spatial and temporal fractional nonlinear convection-diffusion equations

TL;DR

The paper addresses nonlinear space-time fractional convection-diffusion equations with space-fractional order $\beta\in(1,2)$ and time-fractional order $\alpha\in(0,1]$. It develops a local discontinuous Galerkin method using Legendre basis functions by reformulating the FPDE as a first-order system with auxiliary variables and fluxes. A stability analysis yields a convergence bound of $O(h^{k+1}+(\Delta t)^{1+\frac{p}{2}}+p^2)$, and an error estimate is provided under simplifying assumptions, supported by numerical experiments that confirm the theoretical rates. The Legendre-based LDG framework demonstrates enhanced accuracy and robustness for FPDEs across various parameter settings, offering a practical tool for simulations in physics and engineering.

Abstract

This paper focuses on a nonlinear convection-diffusion equation with space and time-fractional Laplacian operators of orders $1<β<2$ and $0<α\leq1$, respectively. We develop local discontinuous Galerkin methods, including Legendre basis functions, for a solution to this class of fractional diffusion problem, and prove stability and optimal order of convergence $O(h^{k+1}+(Δt)^{1+\frac{p}{2}}+p^2)$. This technique turns the equation into a system of first-order equations and approximates the solution by selecting the appropriate basis functions. Regarding accuracy and stability, the basis functions greatly improve the method. According to the numerical results, the proposed scheme performs efficiently and accurately in various conditions and meets the optimal order of convergence.

Theorems & Definitions (15)

• Definition 1
• Lemma 2
• Definition 3
• Lemma 4
• Lemma 5
• proof
• Theorem 6
• proof
• Lemma 7
• proof