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

TL;DR

This work introduces a third-order conservative semi-Lagrangian discontinuous Galerkin method for linear transport on curvilinear unstructured meshes. A novel intersection-based remapping algorithm ensures mass conservation when transferring data between curved upstream elements and the background mesh, enabling accurate, non-splitting, high-order updates augmented by WENO reconstructions and positivity-preserving limiters. The authors establish consistency and mass conservation analytically and validate the method with rigorous numerical experiments, including rigid body rotation and swirling deformations, demonstrating third-order spatial and temporal accuracy and robustness for smooth and discontinuous initial data. The approach significantly extends SLDG capabilities to complex geometries, offering large time-step efficiency and reliable handling of curved geometries in DG frameworks. This has practical implications for multi-tracer transport and climate/atmospheric simulations where conservation and geometric fidelity are critical.

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.

Figures (9)

• Figure 1: Left: space–time region $\widetilde{K}_j(t)\times[t^n,t^{n+1}]$. Middle: nodes on Eulerian element $K_j$ and its upstream element $K_j^\star$. Right: The points $v_q^\star$ on upstream element $K_j^\star$ for least squares approximation and their corresponding points $v_q$ on Eulerian element $K_j$ to evaluate the value of test function $\psi(x,y,t^n)$ on $v_q^\star$, $q=1,2,...,7$.
• Figure 1: Examples for the partition of quadratic curvilinear triangle (left panel) and quadrilateral (right panel), the blue parts represent them in set $"+"$ and the red parts represent them in set $"-"$ in both panels.
• Figure 1: Initial triangular meshes on a circle domain with 25 elements (left) and corresponding upstream meshes with $\Delta t=\frac{\pi}{2}$ of red curves (right).
• Figure 2: Swirling deformation flow. The $L^1$ and $L^2$ errors versus CFL of $P^1$ and $P^2$ SLDG scheme with $T=1$ on a circle domain with the unstructured meshes of 160, 522 and 1884.
• Figure 3: Take third of a cubic triangle (colorful region) as example. The region be divided into six convex domains.

Theorems & Definitions (12)

• Proposition 4.1
• Proof 1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 4.5
• Lemma 4.6
• Lemma 4.7
• Theorem 4.8
• Example 5.1