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A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

Pu Yuan, Paul Zegeling, Xian-Ming Gu

Abstract

In this paper, we study an evolution equation that involves a half-Laplacian operator derived from the Riesz fractional Laplacian, combined with a differential operator \(\mathcal{L}\). Using the identity $(-Δ)^{1/2}=\mathcal H(\partial_x)$, we introduce a Spectrum Doubling (SD) reformulation that transforms the original half-diffusion equation into a first-order doubled system. The reformulated system exhibits stable and unstable spectral branches, and the original half-diffusion dynamics is recovered on a suitable stable invariant subspace characterized by a compatibility condition on the initial condition. The SD reformulation provides a practical numerical advantage: the half-Laplacian is applied only to the initial condition and source term, avoiding repeated evaluation of singular integrals during time marching. For the resulting integer-order system, we develop a Boundary Value Method (BVM) and study a second-order generalized midpoint scheme. We establish its stability and second-order temporal convergence. The fully discrete scheme leads to a large Kronecker-structured linear system, which is solved efficiently by GMRES with a block $ω$-circulant preconditioner. Under simultaneous diagonalizability of the spatial discretization matrices, the preconditioner can be implemented efficiently through fast discrete transforms. Numerical experiments for three evelutionary models confirm the theoretical convergence results and demonstrate the robustness and efficiency of the proposed method, including in strongly advective regimes. The experiments also show that the approach remains effective when the Hilbert transform is evaluated numerically, and illustrate the applicability of the SD framework to a nonlocal Schrödinger-type example.

A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

Abstract

In this paper, we study an evolution equation that involves a half-Laplacian operator derived from the Riesz fractional Laplacian, combined with a differential operator . Using the identity , we introduce a Spectrum Doubling (SD) reformulation that transforms the original half-diffusion equation into a first-order doubled system. The reformulated system exhibits stable and unstable spectral branches, and the original half-diffusion dynamics is recovered on a suitable stable invariant subspace characterized by a compatibility condition on the initial condition. The SD reformulation provides a practical numerical advantage: the half-Laplacian is applied only to the initial condition and source term, avoiding repeated evaluation of singular integrals during time marching. For the resulting integer-order system, we develop a Boundary Value Method (BVM) and study a second-order generalized midpoint scheme. We establish its stability and second-order temporal convergence. The fully discrete scheme leads to a large Kronecker-structured linear system, which is solved efficiently by GMRES with a block -circulant preconditioner. Under simultaneous diagonalizability of the spatial discretization matrices, the preconditioner can be implemented efficiently through fast discrete transforms. Numerical experiments for three evelutionary models confirm the theoretical convergence results and demonstrate the robustness and efficiency of the proposed method, including in strongly advective regimes. The experiments also show that the approach remains effective when the Hilbert transform is evaluated numerically, and illustrate the applicability of the SD framework to a nonlocal Schrödinger-type example.

Paper Structure

This paper contains 36 sections, 7 theorems, 171 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The half-Laplacian $(-\Delta)^{\frac{1}{2}}$
  4. The Hilbert Transform H
  5. Spectrum Doubling Reformulation
  6. Doubled spectral system and the stable invariant subspace
  7. Second-order abstract Cauchy problem
  8. Stable invariant subspace and induced contraction semigroup
  9. Consistency between (\ref{['original_half']}) and (\ref{['ds_eq']})
  10. Explicit Solution of (\ref{['ds_eq']})
  11. Equivalence of Solutions
  12. Case 1: $\mathcal{L}=0$
  13. Case 2: $\mathcal{L} = \delta\mathcal{I}$
  14. Case 3: $\mathcal{L} = \delta\partial_x$
  15. The time-integration method
  16. ...and 21 more sections

Key Result

Lemma 3.1

Let the initial state $(u(0),v(0))=(u_0,v_0)$. Under the regular assumption of $u$, if the initial state $(u_0,v_0)\in\mathbb{M}_s$, then we have $(u,v)\in \mathbb{M}_s$ for $t\in[0,T]$.

Figures (12)

  • Figure 1: Stability (boundary) locus of the method of explicit$\slash$implicit Euler, RK2, RK4, BDF2, BDF4 and RADAU IIA.
  • Figure 2: Numerical solution of the half-diffusion heat model (\ref{['heat']}) with $f(x,t) = 0$, initialized with $u_0(x) = \frac{1}{(1+x^2)^2}$ and homogeneous Dirichlet boundary condition, with $\tau=0.0391$ and $h=0.0195$ at time $T=20$.
  • Figure 3: Comparison at $T=20$ for (\ref{['heat']}). Left: relative $L^2$-errors versus $\tau$ for the preconditioned BVM (solid lines) and the standard BVM (dashed lines) with different $h=\frac{L}{128},\frac{L}{256},\frac{L}{512},\frac{L}{1024},\frac{L}{2048}$. The black line denotes the reference second-order convergence rate. Right: speedup ratio for the same spatial step sizes $h$.
  • Figure 4: CPU time of GMRES with the preconditioner (solid lines) and without the preconditioner (dashed lines) for different $h=\frac{L}{128},\frac{L}{256},\frac{L}{512},\frac{L}{1024}$.
  • Figure 5: The numerical solution of (\ref{['diffusion_u']}) with $\epsilon=0.1,\delta=0.02$, initialized with $u_0(x) = \frac{1}{(1+x^2)^2}$ and homogeneous Dirichlet boundary condition, with $\tau=0.0391$ and $h=0.0195$ at time $T=20$.
  • ...and 7 more figures

Theorems & Definitions (22)

  • Remark
  • Definition 1
  • Definition 2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark
  • Theorem 3.3
  • Remark
  • ...and 12 more