A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

TL;DR

This paper develops a high-order semi-Lagrangian IMEX discontinuous Galerkin method on 2D staggered unstructured meshes for advection-diffusion, incompressible Navier–Stokes, and natural convection. By placing velocity on a dual edge-based grid and scalars on the main grid, convection is treated explicitly via semi-Lagrangian trajectories while diffusion and pressure are handled implicitly, with IMEX Runge-Kutta time stepping achieving up to third order in time. A key innovation is the edge-based dual basis formed by splitting dual cells into two sub-triangles, enabling reference-operator reduction and a symmetric positive-definite pressure system that can be solved efficiently with a matrix-free conjugate gradient method; this yields unconditional stability and reduced memory for the DG operators. The framework is validated across nonlinear transport, advection–diffusion, Taylor-Green vortex, and density-current problems, showing robust high-order accuracy and efficiency that scales to complex fluid-thermal applications on unstructured meshes.

Abstract

In this paper we present a new high order semi-implicit DG scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

Figures (11)

• Figure 1: Example of a triangular element $\mathbf{T}_i$ composed by the edges $\{\Gamma_{j_1},\Gamma_{j_2},\Gamma_{j_3}\}$ with its neighbors. The quadrilateral dual element $\mathbf{R}_{j_1}$ is composed by the union of the left and right sub-triangles adjacent to the edge $\Gamma_{j_1}$, namely $\ell(j_1)=i$ and $r(j_1)=i_1$.
• Figure 2: Example of functional space on the standard square $R_{ref}$ for $p=2$ with $9$ degrees of freedom (cross symbols). Tensor product of one dimensional basis functions (left) and reinterpretation in the context of agglomerated elements by split into two sub-triangles (right).
• Figure 3: Schematic of the reference maps. All the possible combinations are three for $T_\alpha$ and two for $T_\beta$
• Figure 4: Two dimensional solution of non-linear transport test: comparison between exact and numerical solution with $R=1,2$ and extraction slices at $x=0.25$ and $y=0$.
• Figure 5: Non-linear transport. Comparison of the numerical solution obtained with $\Delta t=1.0$ and the exact one on the slice $y=0$ (left) and $x=0.25$ (right).