Fourth- and Higher-Order Semi-Lagrangian Finite Volume Methods for the Two-dimensional Advection Equation on Arbitrarily Complex Domains

TL;DR

This paper introduces a family of high-order semi-Lagrangian finite volume methods for solving the 2D advection equation, capable of handling complex domains with high accuracy and robustness.

Contribution

The paper presents novel fourth- to eighth-order semi-Lagrangian finite volume methods that work on complex geometries and are easy to implement for various boundary conditions.

Findings

Achieved fourth- to eighth-order convergence rates.

Demonstrated effectiveness on irregular and complex domains.

Confirmed robustness and excellent conditioning of the methods.

Abstract

To numerically solve the two-dimensional advection equation, we propose a family of fourth- and higher-order semi-Lagrangian finite volume (SLFV) methods that feature (1) fourth-, sixth-, and eighth-order convergence rates, (2) applicability to both regular and irregular domains with arbitrarily complex topology and geometry, (3) ease of handling both zero and nonzero source terms, and (4) the same algorithmic steps for both periodic and incoming penetration conditions. Test results confirm the analysis and demonstrate the accuracy, flexibility, robustness, and excellent conditioning of the proposed SLFV method.