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Pointwise-in-time convergence analysis of an Alikhanov scheme for a 2D nonlinear subdiffusion equation

Chang Hou, Hu Chen, Jian Wang

TL;DR

The paper develops a linearized, fully discrete Alikhanov scheme on a quasi-graded temporal mesh to solve a nonlinear 2D time-fractional subdiffusion equation with Caputo derivatives. A new stability result enables robust pointwise (in time) error analysis, yielding global L^2 convergence of order min{α r, 2} and local L^2 convergence of order min{r, 2}, with achieving 2nd-order accuracy away from t = 0 for sufficiently graded meshes. The analysis combines truncation error bounds, a barrier-based stability framework, and a Newton linearization of the nonlinear term, producing sharp, α-dependent convergence rates. Numerical experiments on nonlinear and linear cases corroborate the theoretical rates, demonstrating the method’s effectiveness on nonuniform temporal meshes. The results are relevant for accurate, stable simulation of nonlinear subdiffusion in two spatial dimensions on graded time grids.

Abstract

In this paper, we discretize the Caputo time derivative of order α\in (0,1) using the Alikhanov scheme on a quasi-graded temporal mesh, and employ the Newton linearization method to approximate the nonlinear term. This yields a linearized fully discrete scheme for the two-dimensional nonlinear time fractional subdiffusion equation with weakly singular solutions. For the purpose of conducting a pointwise convergence analysis using the comparison principle, we develop a new stability result. The global L^2-norm convergence order is min{αr, 2}, and the local L^2-norm convergence order is min{r, 2} under appropriate conditions and assumptions. Ultimately, the rates of convergence demonstrated by the numerical experiments serve to validate the analytical outcomes.

Pointwise-in-time convergence analysis of an Alikhanov scheme for a 2D nonlinear subdiffusion equation

TL;DR

The paper develops a linearized, fully discrete Alikhanov scheme on a quasi-graded temporal mesh to solve a nonlinear 2D time-fractional subdiffusion equation with Caputo derivatives. A new stability result enables robust pointwise (in time) error analysis, yielding global L^2 convergence of order min{α r, 2} and local L^2 convergence of order min{r, 2}, with achieving 2nd-order accuracy away from t = 0 for sufficiently graded meshes. The analysis combines truncation error bounds, a barrier-based stability framework, and a Newton linearization of the nonlinear term, producing sharp, α-dependent convergence rates. Numerical experiments on nonlinear and linear cases corroborate the theoretical rates, demonstrating the method’s effectiveness on nonuniform temporal meshes. The results are relevant for accurate, stable simulation of nonlinear subdiffusion in two spatial dimensions on graded time grids.

Abstract

In this paper, we discretize the Caputo time derivative of order α\in (0,1) using the Alikhanov scheme on a quasi-graded temporal mesh, and employ the Newton linearization method to approximate the nonlinear term. This yields a linearized fully discrete scheme for the two-dimensional nonlinear time fractional subdiffusion equation with weakly singular solutions. For the purpose of conducting a pointwise convergence analysis using the comparison principle, we develop a new stability result. The global L^2-norm convergence order is min{αr, 2}, and the local L^2-norm convergence order is min{r, 2} under appropriate conditions and assumptions. Ultimately, the rates of convergence demonstrated by the numerical experiments serve to validate the analytical outcomes.
Paper Structure (6 sections, 17 theorems, 80 equations, 6 tables)

This paper contains 6 sections, 17 theorems, 80 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Fully discrete scheme on a quasi-graded mesh
  3. Truncation error estimates
  4. Stability result for Alikhanov scheme
  5. Convergence and stability analysis
  6. Numerical examples

Key Result

Lemma 2.1

MR3936261$\delta_t^{\alpha,*}U^m$ can be represented as $\delta_t^{\alpha,*}U^m=\sum_{j=1}^{m}{g_{m-1,j-1}(U^j-U^{j-1})}$ as well. The coefficient $g_{0,0}=\tau_1^{-1}a_{0,0}$ and when $k\ge 1$ coefficients with $a_{k,k}=\frac{\sigma^{1-\alpha}}{\Gamma(2-\alpha)}\tau_{k+1}^{1-\alpha}$ for $k\ge0$, $a_{k,j}=\frac{1}{\Gamma(1-\alpha)}\int_{t_j}^{t_{j+1}}{(t_{k+1}^*-\eta)^{-\alpha}d\eta}$, $b_{k,j}=

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 20 more