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 \(\mathcal{L}\). Using the identity $(-\Delta)^{1/2}=\mathcal H(\partial_x)$, 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 $\omega$-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 Schr\"odinger-type example.