Unifying Finite Differences and Semi-Lagrangian Schemes via Localized Matrix Exponentials
This work addresses a foundational problem in computational physics by providing a more efficient and accurate method for solving partial differential equations, though it appears incremental as it builds on existing exponential integrator techniques.
The paper tackled the trade-off between accuracy and efficiency in numerical methods by introducing a unified framework for localized exponential integrators, which achieved high-order temporal accuracy and superior stability at high Courant numbers, with optimal O(N) scaling and execution times independent of spatial order in experiments.
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) $Ï$-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.