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A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation

Yong-Liang Zhao, Pei-Yong Zhu, Xian-Ming Gu, Xi-Le Zhao, Huan-Yan Jian

TL;DR

This work tackles the all-at-once system arising from a nonlinear tempered fractional diffusion equation with variable coefficients. It develops two implicit schemes, NL-IES and L-IES, proving their stability and first-order convergence in time and space, and then formulates a nonlinear all-at-once system solved by Newton's method with a coarse-grid initialization. To accelerate the Newton iterations, a robust preconditioner $P_\ell$ is designed, with a block-tridiagonal structure and FFT-enabled matrix-vector products that yield $\mathcal{O}(MN\log MN)$ per iteration and well-clustered eigenvalues for the preconditioned Jacobian. Numerical experiments on continuous and discontinuous coefficient problems demonstrate the expected first-order accuracy and substantial reductions in iteration counts and compute time when using the preconditioner. The results suggest strong potential for parallelization of the all-at-once approach in NL-TFDEs, with further refinements anticipated to enhance preconditioner effectiveness.

Abstract

An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may suitable for parallel computations. Newton's method, whose initial value is obtained by interpolating the solution of the linearized implicit scheme on the coarse space, is chosen to solve such the nonlinear all-at-once system. To accelerate the speed of solving the Jacobian equations appeared in Newton's method, a robust preconditioner is developed and analyzed. Numerical examples are reported to demonstrate the effectiveness of our proposed preconditioner. Meanwhile, they also imply that such the initial guess for Newton's method is more suitable.

A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation

TL;DR

This work tackles the all-at-once system arising from a nonlinear tempered fractional diffusion equation with variable coefficients. It develops two implicit schemes, NL-IES and L-IES, proving their stability and first-order convergence in time and space, and then formulates a nonlinear all-at-once system solved by Newton's method with a coarse-grid initialization. To accelerate the Newton iterations, a robust preconditioner is designed, with a block-tridiagonal structure and FFT-enabled matrix-vector products that yield per iteration and well-clustered eigenvalues for the preconditioned Jacobian. Numerical experiments on continuous and discontinuous coefficient problems demonstrate the expected first-order accuracy and substantial reductions in iteration counts and compute time when using the preconditioner. The results suggest strong potential for parallelization of the all-at-once approach in NL-TFDEs, with further refinements anticipated to enhance preconditioner effectiveness.

Abstract

An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may suitable for parallel computations. Newton's method, whose initial value is obtained by interpolating the solution of the linearized implicit scheme on the coarse space, is chosen to solve such the nonlinear all-at-once system. To accelerate the speed of solving the Jacobian equations appeared in Newton's method, a robust preconditioner is developed and analyzed. Numerical examples are reported to demonstrate the effectiveness of our proposed preconditioner. Meanwhile, they also imply that such the initial guess for Newton's method is more suitable.

Paper Structure

This paper contains 10 sections, 7 theorems, 44 equations, 2 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two implicit schemes and all-at-once system
  3. Two implicit schemes
  4. The all-at-once system
  5. Stabilities and convergences of (2.3) and (2.4)
  6. The stabilities of (2.3) and (2.4)
  7. The convergences of (2.3) and (2.4)
  8. The preconditioned iterative method
  9. Numerical examples
  10. Concluding remarks

Key Result

Lemma 3.1

(sabzikar2015tempered ) The coefficients $g_k^{(\alpha)}$, for $k = 0, 1, \cdots$, satisfy:

Figures (2)

  • Figure 1: Spectra of $J^{0}$ and $P_{\ell}^{-1} J^{0}$, when $\alpha = 1.5$, $M = N = 65$ in Example 1. Top row: $\lambda = 0$; Bottom row: $\lambda = 5$.
  • Figure 2: Spectra of $J^{0}$ and $P_{\ell}^{-1} J^{0}$, when $\alpha = 1.9$, $M = N = 65$ in Example 2. Top row: $\lambda = 0$; Bottom row: $\lambda = 10$.

Theorems & Definitions (7)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Corollary 1
  • Theorem 3.2
  • Corollary 2
  • Theorem 4.1