Lot-Kei Chou | Searcharxiv

arXiv2025

Upwind-and-shifted numerical scheme for fractional convection equation

Fundamental solution of a space fractional convection equation of order $α$ is the probability density function of Lévy flights with long-tailed $α$-stable jump length distribution. By studying an upwind second-order implicit finite difference scheme for the equation with $α\in(0,1)$, an upwind-and-shifted scheme with order $3-α$ is obtained in this paper, and the scheme is shown to be unconditionally stable for a wide range of $α$. Numerical examples, including simulations on a probability density function, are presented showing the effectiveness of the numerical schemes.

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arXiv2024

Optimal preconditioners for nonsymmetric multilevel Toeplitz systems with application to solving non-local evolutionary partial differential equations

Preconditioning for multilevel Toeplitz systems has long been a focal point of research in numerical linear algebra. In this work, we develop a novel preconditioning method for a class of nonsymmetric multilevel Toeplitz systems, which includes the all-at-once systems that arise from evolutionary partial differential equations. These systems have recently garnered considerable attention in the literature. To further illustrate our proposed preconditioning strategy, we specifically consider the application of solving a wide range of non-local, time-dependent partial differential equations in a parallel-in-time manner. For these equations, we propose a symmetric positive definite multilevel Tau preconditioner that is not only efficient to implement but can also be adapted as an optimal preconditioner. In this context, the proposed preconditioner is optimal in the sense that it enables mesh-independent convergence when using the preconditioned generalized minimal residual method. Numerical examples are provided to critically analyze the results and underscore the effectiveness of our preconditioning strategy.

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