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Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

Yi Yang, Jin Huang

Abstract

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our analysis shows that the choice of graded mesh factor $ω=(2-α)/α$ shall give an optimal temporal convergence $\mathcal{O}(N^{-(2-α)})$ with $N$ denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

Abstract

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our analysis shows that the choice of graded mesh factor shall give an optimal temporal convergence with denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.
Paper Structure (11 sections, 11 theorems, 129 equations, 2 figures, 7 tables)

This paper contains 11 sections, 11 theorems, 129 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Spatial discretization
  3. Matrix transform technique
  4. Spatial semi-discretization
  5. Implementation and analysis of full discrete scheme
  6. Fast time-stepping method
  7. Convergence analysis
  8. Numerical experiments
  9. Accuracy test
  10. Time-dependent problems
  11. Conclusion

Key Result

Lemma 3.1

Let $0<\alpha<1$, $\beta \in \mathbb{R}$, and $\frac{\alpha \pi}{2}<\mu<\pi\alpha$. Then there exists constant $C=C\left(\alpha, \beta, \mu\right)>0$ such that

Figures (2)

  • Figure 1: Process of the evolution of time-space fractional Allen-Cahn equation with the kissing bubbles: $\alpha=0.4$, $0.6$, $0.8$, $1$ from left to right, respectively.
  • Figure 2: Snapshots of the numerical solutions of time-space fractional Cahn-Hilliard equation with $\beta=1.0$, $0.8$, $0.6$, $0.4$ from left to right, respectively.

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.1
  • ...and 15 more