Unifying Finite Differences and Semi-Lagrangian Schemes via Localized Matrix Exponentials

TL;DR

This paper introduces a unified, localized exponential integrator framework that combines the accuracy of spectral methods with the efficiency of finite differences, achieving high-order accuracy and stability for complex PDEs.

Contribution

The authors develop a Local Matrix Exponential Propagator (LMEP) that unifies finite difference and semi-Lagrangian schemes, enabling high-order, stable, and efficient time integration for PDEs.

Findings

Preserves high-order temporal accuracy.

Exhibits superior stability at high Courant numbers.

Achieves optimal linear scaling with problem size.

Abstract

We present a unified framework for the construction of localized exponential integrators that bypasses the traditional trade-off between the accuracy of global spectral methods and the efficiency of sparse finite differences. By evaluating the matrix exponential of a discrete operator strictly within a local stencil of size $n$, we "harvest" integration weights that naturally incorporate high-order temporal corrections. We prove that this Local Matrix Exponential Propagator (LMEP) is algebraically isomorphic to optimal semi-Lagrangian transport for advection and provides algebraically exact coupled evolution for mixed-physics operators, effectively eliminating the commutator errors associated with operator splitting. The framework is extended to semi-linear systems via a localized augmented matrix approach, facilitating the evaluation of Exponential Time Differencing (ETD) $\phi$-functions through sparse, banded operations. Numerical experiments on the viscous Burgers, Korteweg-de Vries, and Allen-Cahn equations demonstrate that the method preserves high-order temporal accuracy and exhibits superior stability at high Courant numbers across both periodic and non-periodic domains. We empirically demonstrate that this localized approach yields optimal $\mathcal{O}(N)$ scaling and, for high-CFL upwind configurations, total execution times that remain strictly independent of the spatial approximation order.