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

Víctor Bayona

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) $φ$-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.

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

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 , 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) -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 scaling and, for high-CFL upwind configurations, total execution times that remain strictly independent of the spatial approximation order.
Paper Structure (15 sections, 2 theorems, 16 equations, 5 figures)

This paper contains 15 sections, 2 theorems, 16 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Unified Generator Framework
  3. Local Matrix Exponential Propagator
  4. Advection and Algebraic Isomorphism
  5. Generalization to Diffusive and Mixed Operators
  6. Generalization to Semi-Linear PDEs
  7. The Augmented LMEP Framework
  8. Numerical Validations
  9. High-CFL Transport and Spectral Semi-Lagrangian Isomorphism
  10. Exact Coupled Evolution: Bypassing Operator Splitting
  11. Non-Linear PDEs
  12. Burgers' Equation
  13. Korteweg-de Vries (KdV) Equation
  14. Allen-Cahn Equation
  15. Conclusion

Key Result

Theorem 1

Let $\mathbf{u} \in \mathbb{R}^n$ represent the values of a polynomial $p(x) \in \mathcal{P}_{n-1}$ on the stencil. The vector $\mathbf{v} = \exp(-a \Delta t \mathbf{D}_n)\mathbf{u}$ contains the exact values of the shifted polynomial $p(x - a\Delta t)$. Consequently, the weights of the LMEP are ide

Figures (5)

  • Figure 1: Numerical Analysis of the Localized ETD Operator. Top: convergence reaching machine precision for Lagrangian transport. Middle/Bottom: spectral breakdown (dispersion and diffusion) for non-Lagrangian configurations. The one-sided configuration permits extremely high CFL conditions ($\nu \gg 1$) but encounters stability limits at $n=25$ due to edge-point sensitivity. The included efficiency data highlights that total execution time remains flat as $n$ increases, owing to the $\Delta t \propto n$ scaling.
  • Figure 2: Error convergence landscapes for $N \in \{64, 128, 512\}$. The black line indicates the stability boundary $\Delta t^*$, showing the expansion of the stable operating envelope as a function of $n$. All plots share the same logarithmic scale to highlight the spectral convergence floor.
  • Figure 3: Numerical analysis for the viscous Burgers' equation with $\nu = 0.03$ and $n=19$. (a) Relative $\ell_\infty$ error convergence, demonstrating that the localized ETD-RK4 scheme preserves fourth-order accuracy in time ($\mathcal{O}(\Delta t^4)$) until the spatial error limit is reached. (b) Computational complexity, showing the total execution time scaling linearly ($\mathcal{O}(N)$) with grid resolution across five different time steps ($\Delta t$). This confirms the optimal efficiency of the sparse, banded operator implementation.
  • Figure 4: Numerical results for the KdV equation using parameters from Kassam2005. Top: Space-time evolution $u(x,t)$ showing the two-soliton interaction and phase shift using LETD. Bottom-left: Comparison of mass conservation error ($\ell_1$ norm). Bottom-right: Comparison of energy conservation error ($\ell_2$ squared). The localized ETD method ($n=23$) demonstrates accuracy and conservation properties indistinguishable from the global spectral reference.
  • Figure 5: Phase dynamics of the Allen--Cahn equation ($\epsilon = 0.01$) on a Chebyshev grid ($N=64$). The evolution captures the formation of stable equilibria and the phenomenon of metastability. Initial conditions and comparison benchmarks are adapted from Kassam2005. The localized ETD method tracks the $N=1024$ global spectral reference with a final $\ell_\infty$ error of $10^{-5}$.

Theorems & Definitions (9)

  • Theorem 1: Isomorphism to Lagrange Interpolation
  • proof
  • Example 1: Automatic Generation of Advective Schemes
  • Remark 1: Exactness via Nilpotency
  • Theorem 2: General Exactness
  • proof
  • Remark 2: No Operator Splitting
  • Example 2: Automatic Generation of Diffusive Schemes
  • Remark 3: Stability and Stencil Size