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Numerical analysis for subdiffusion problem with non-positive memory

Wenlin Qiu, Xiangcheng Zheng

TL;DR

The paper addresses numerical schemes for subdiffusion equations with non-positive memory kernels, deriving a time discretization based on a nonuniform $L1$ method for the Caputo derivative and an interpolation quadrature for the memory term. It introduces complementary discrete convolution kernels to establish stability and first-order temporal accuracy, and extends existing analyses to the subdiffusive regime with non-positive memory. A fully discrete Galerkin scheme is analyzed, yielding stability and an overall error bound of $O(N^{-1}+h^2)$ under a graded mesh condition $\gamma \ge 1/\sigma$. Numerical experiments on positive-type, tempered, and non-positive kernels confirm the expected first-order time and second-order space convergence, demonstrating the method’s robustness for various memory behaviors and dimensions.

Abstract

This work considers the subdiffusion problem with non-positive memory, which not only arises from physical laws with memory, but could be transformed from sophisticated models such as subdiffusion or subdiffusive Fokker-Planck equation with variable exponent. We apply the non-uniform L1 formula and interpolation quadrature to discretize the fractional derivative and the memory term, respectively, and then adopt the complementary discrete convolution kernel approach to prove the stability and first-order temporal accuracy of the scheme. The main difficulty in numerical analysis lies in the non-positivity of the kernel and its coupling with the complementary discrete convolution kernel (such that different model exponents are also coupled), and the results extend those in [Chen, Thomée and Wahlbin, Math. Comp. 1992] to the subdiffusive case. Numerical experiments are performed to substantiate the theoretical results.

Numerical analysis for subdiffusion problem with non-positive memory

TL;DR

The paper addresses numerical schemes for subdiffusion equations with non-positive memory kernels, deriving a time discretization based on a nonuniform method for the Caputo derivative and an interpolation quadrature for the memory term. It introduces complementary discrete convolution kernels to establish stability and first-order temporal accuracy, and extends existing analyses to the subdiffusive regime with non-positive memory. A fully discrete Galerkin scheme is analyzed, yielding stability and an overall error bound of under a graded mesh condition . Numerical experiments on positive-type, tempered, and non-positive kernels confirm the expected first-order time and second-order space convergence, demonstrating the method’s robustness for various memory behaviors and dimensions.

Abstract

This work considers the subdiffusion problem with non-positive memory, which not only arises from physical laws with memory, but could be transformed from sophisticated models such as subdiffusion or subdiffusive Fokker-Planck equation with variable exponent. We apply the non-uniform L1 formula and interpolation quadrature to discretize the fractional derivative and the memory term, respectively, and then adopt the complementary discrete convolution kernel approach to prove the stability and first-order temporal accuracy of the scheme. The main difficulty in numerical analysis lies in the non-positivity of the kernel and its coupling with the complementary discrete convolution kernel (such that different model exponents are also coupled), and the results extend those in [Chen, Thomée and Wahlbin, Math. Comp. 1992] to the subdiffusive case. Numerical experiments are performed to substantiate the theoretical results.
Paper Structure (10 sections, 6 theorems, 88 equations, 6 tables)

This paper contains 10 sections, 6 theorems, 88 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Main contribution
  4. Time-discrete scheme
  5. Numerical scheme
  6. Stability analysis
  7. Convergence estimate
  8. Fully discrete Galerkin scheme
  9. Numerical experiment
  10. Conclusion

Key Result

lemma thmcounterlemma

LiaoLiao1 The complementary discrete convolution kernel $P_{n-k}^{(n)}$ satisfies the following properties: (i) $0< P_{n-k}^{(n)} \leq \Gamma(2-\sigma) \tau_k^{\sigma}$ for $1\leq k \leq n$; (ii) $\sum_{j=k}^{n}P_{n-k}^{(n)}a_{j-k}^{(j)}=1$ for $1\leq k \leq n$; (iii) For $q=0,1$ it holds

Theorems & Definitions (10)

  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof