A Third-order Conservative Semi-Lagrangian Discontinuous Galerkin Scheme For the Transport Equation on Curvilinear Unstructured Meshes

TL;DR

This paper introduces a third-order conservative semi-Lagrangian discontinuous Galerkin scheme for linear transport equations on complex curvilinear unstructured meshes, combining high accuracy, mass conservation, and stability for complex geometries.

Contribution

It develops a high-order conservative remapping algorithm and integrates it into a non-splitting SLDG method with limiters, enabling accurate, stable, and efficient solutions on unstructured meshes.

Findings

Achieves third-order accuracy in space and time.

Effectively suppresses numerical oscillations and maintains positivity.

Validated through benchmarks demonstrating stability and robustness.

Abstract

We develop a third-order conservative semi-Lagrangian discontinuous Galerkin (SLDG) scheme for solving linear transport equations on curvilinear unstructured triangular meshes, tailored for complex geometries. To ensure third-order spatial accuracy while strictly preserving mass, we develop a high-order conservative intersection-based remapping algorithm for curvilinear unstructured meshes, which enables accurate and conservative data transfer between distinct curvilinear meshes. Incorporating this algorithm, we construct a non-splitting high-order SLDG method equipped with weighted essentially non-oscillatory and positivity-preserving limiters to effectively suppress numerical oscillations and maintain solution positivity. For the linear problem, the semi-Lagrangian update enables large time stepping, yielding an explicit and efficient implementation. Rigorous numerical analysis confirms that our scheme achieves third-order accuracy in both space and time, as validated by consistent error analysis in terms of $L^1$ and $L^2$-norms. Numerical benchmarks, including rigid body rotation and swirling deformation flows with smooth and discontinuous initial conditions, validate the scheme's accuracy, stability, and robustness.