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

Xiaofeng Cai; Yibing Chen; Kunkai Fu; Liujun Pan

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arXiv · 2511.10100

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

Xiaofeng Cai·Yibing Chen·Kunkai Fu·Liujun Pan

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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.

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Xiaofeng Cai, Yibing Chen, Kunkai Fu, Liujun Pan. 2025-11-13. A Third-order Conservative Semi-Lagrangian Discontinuous Galerkin Scheme For the Transport Equation on Curvilinear Unstructured Meshes. https://arxiv.org/abs/2511.10100

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