In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where the Euler equations have been solved solved using a semi-implicit finite volume scheme based on the flux-vector splitting method recently proposed by Toro and Vázquez-Cendón (Comput Fluids 70:1–12, 2012). In our scheme, the nonlinear convective terms are discretized explicitly, while the pressure terms are discretized implicitly. As a consequence, the time step is restricted only by a mild CFL condition based on the fluid velocity, which makes this method particularly suitable for simulations in the low Mach number regime. However, the conservative formulation of the scheme, together with the novel subcell finite volume limiter allows also the numerical simulation of high Mach number flows with shock waves. Inserting the discrete momentum equation into the discrete total energy conservation law yields a mildly nonlinear system with the scalar pressure as the only unknown. Due to the use of staggered meshes, the resulting pressure system has the most compact stencil possible and can be efficiently solved with modern iterative methods. In order to deal with shock waves or steep gradients, the new semi-implicit DG scheme proposed in this paper includes an a posteriori subcell finite volume limiting technique. This strategy was first proposed by Dumbser et al. (J Comput Phys 278:47–75, 2014) for explicit DG schemes on collocated grids and is based on the a posteriori MOOD algorithm of Clain, Loubère and Diot. Recently, this methodology was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in Ioriatti and Dumbser (Appl Numer Math 135:443–480, 2019). Within the MOOD approach, an unlimited DG scheme first produces a so-called candidate solution for the next time level \(t^{n+1}\). Later on, the control volumes with a non-admissible candidate solution are identified by using physical and numerical detection criteria, such as the positivity of the solution, the absence of floating point errors and the satisfaction of a relaxed discrete maximum principle (DMP). Then, in the detected troubled cells a more robust first order semi-implicit finite volume (FV) method is applied on a sub-grid composed of \(2P + 1\) subcells, where P denotes the polynomial degree used in the DG scheme. For that purpose, the nonlinear convective terms are recomputed in the troubled cells using an explicit finite volume scheme on the subcell level. Also the linear system for the pressure needs to be assembled and solved again, but where now a low order semi-implicit finite volume scheme is used on the sub-cell level in all troubled DG elements, instead of the original high order DG method. Finally, the higher order DG polynomials are reconstructed from the piecewise constant subcell finite volume averages and the scheme proceeds to the next time step. In this paper we present, discuss and test this novel family of methods and simulate a set of classical numerical benchmark problems of compressible gasdynamics. Great attention is dedicated to 1D and 2D Riemann problems and we also show that for these test cases the scheme responds appropriately in the presence of shock waves and does not produce non-physical spurious numerical oscillations.

在本文中，我们为一维和二维可压缩气体动力学欧拉方程，提出了一种具有后验子单元有限体积限制的交错网格上的新型半隐式不连续Galerkin（DG）有限元方案。因此，我们扩展了Dumbser和Casulli（Appl Math Comput 272：479–497，2016）所采用的策略，其中基于基于最近提出的磁通矢量分裂方法的半隐式有限体积方案，求解了Euler方程。 Toro和Vázquez-Cendón（计算流体70：1–12，2012年）。在我们的方案中，非线性对流项被显式离散，而压力项被隐式离散。因此，时间步长仅受基于流体速度的温和CFL条件限制，这使得该方法特别适合在低马赫数条件下进行仿真。但是，该方案的保守公式与新颖的子单元有限体积限制器一起，还可以对带有冲击波的高马赫数流进行数值模拟。将离散动量方程式插入到离散总能量守恒定律中，将产生一个温和非线性的系统，其中标量压力是唯一未知的系统。由于使用了交错的网格，因此生成的压力系统具有尽可能紧凑的模板，并且可以使用现代的迭代方法有效地解决。为了处理冲击波或陡峭的梯度，本文提出的新的半隐式DG方案包括后验子单元有限体积限制技术。这种策略最初是由Dumbser等人提出的。（J Comput Phys 278：47-75，2014），用于并置网格上的显式DG方案，它基于Clain，Loubère和Diot的后验MOOD算法。最近，这种方法还扩展到了Ioriatti和Dumbser中浅水方程组的交错网格上的半隐式DG方案（Appl Numer Math 135：443–480，2019）。在MOOD方法中，无限制DG方案首先为下一个时间级别\（t ^ {n + 1} \）生成所谓的候选解决方案。后来，通过使用物理和数字检测标准来识别具有不允许的候选解决方案的控制体积，例如，解决方案的阳性，不存在浮点误差以及满足松弛离散最大原理（DMP）。然后，在检测到的故障单元中，将更鲁棒的一阶半隐式有限体积（FV）方法应用于由\（2P + 1 \）个子单元组成的子网格，其中P表示DG方案中使用的多项式次数。为此，使用子单元级上的显式有限体积方案在有问题的单元中重新计算非线性对流项。同样，需要重新组装和求解线性压力系统，但是现在在所有有问题的DG元件的子电池层上都使用了低阶半隐式有限体积方案，而不是原来的高阶DG方法。最后，从分段常数子单元有限体积平均值中重建高阶DG多项式，然后该方案进行到下一个时间步。在本文中，我们介绍，讨论和测试了这一新颖的方法系列，并模拟了一组可压缩气体动力学的经典数值基准问题。