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.

翻译：本文提出了一种应用于二维交错三角形网格的高阶半隐式DG方法，用于求解不同非线性双曲守恒律系统，如对流-扩散模型、不可压缩纳维-斯托克斯方程及自然对流问题。其中，温度场和压力场定义在三角形主网格上，而速度场定义在基于四边形边的交错网格上。我们提出了一种半隐式时间离散方法，通过显式和隐式处理分别分离慢速和快速时间尺度。非线性对流项采用半拉格朗日方法显式演化，而扩散项和压力贡献则采用隐式离散。时间方向的高阶精度通过一种新的灵活通用框架实现，该框架专门设计用于与半拉格朗日方法配合的IMplicit-EXplicit (IMEX) 龙格-库塔格式。为提高DG散度算子和质量矩阵的计算效率，我们建议在基于边的网格上使用正则性较低的多项式空间来近似数值解，该网格定义在分割交错四边形元素的两个子三角形上。由于对快速尺度项进行了隐式处理，所得数值格式对所考虑的控制方程是无条件稳定的。与真正的时空间断伽辽金格式不同，IMEX离散能够保持所生成的压力线性系统的对称性和半正定性，并可通过高效的免矩阵共轭梯度方法求解。我们展示了几项收敛性结果，包括非线性输运和密度流，在空间和时间上均达到三阶精度。