Analysis of an Explicit, High-Order Semi-Lagrangian Nodal Method

TL;DR

This paper provides a detailed analysis of an explicit semi-Lagrangian spectral element method, examining its phase and dissipation errors, stability, and consistency with the transport PDE, using algebraic, modified equation, and eigenvalue analyses.

Contribution

It introduces a discrete algebraic formulation of the high-order semi-Lagrangian method and analyzes its error characteristics and stability properties.

Findings

Method is consistent with PDE as element size approaches zero.

Dispersion is negligible, indicating high accuracy.

Stable under CFL condition based on minimum node spacing.

Abstract

A discrete analysis of the phase and dissipation errors of an explicit, semi-Lagrangian spectral element method is performed. The semi-Lagrangian method advects the Lagrange interpolant according the Lagrangian form of the transport equations and uses a least-square fit to correct the update for interface constraints of neighbouring elements. By assuming a monomial representation instead of the Lagrange form, a discrete version of the algorithm on a single element is derived. The resulting algebraic system lends itself to both a Modified Equation analysis and an eigenvalue analysis. The Modified Equation analysis, which Taylor expands the stencil at a single space location and time instance, shows that the semi-Lagrangian method is consistent with the PDE form of the transport equation in the limit that the element size goes to zero. The leading order truncation term of the Modified Equation is of the order of the degree of the interpolant which is consistent with numerical tests reported in the literature. The dispersion relations show that the method is negligibly dispersive, as is common for semi-Lagrangian methods. An eigenvalue analysis shows that the semi-Lagrangian method with a nodal Chebyshev interpolant is stable for a Courant-Friedrichs-Lewy condition based on the minimum collocation node spacing within an element that is greater than unity.