Communications in Applied Mathematics and Computational Science ( IF 2.1 ) Pub Date : 2021-01-19 , DOI: 10.2140/camcos.2021.16.59 Dinshaw S. Balsara , Rakesh Kumar , Praveen Chandrashekar
The high-accuracy solution of the MHD equations is of great interest in various fields of physics, mathematics, and engineering. Higher-order DG schemes offer low dissipation and dispersion as well as the ability to model complex geometries, which is very desirable in various applications. Numerical solution of the MHD equations is made challenging by the fact that the PDE system has an involution constraint. Therefore, we construct high-order, globally divergence-free DG schemes for compressible MHD. The modes of the fluid variables are collocated at the zones of the mesh; the magnetic field components and their higher-order modes are collocated at the faces of the mesh. The fluid equations are evolved using classical DG, while the magnetic fields are evolved using a novel DG-like approach, first proposed by Balsara and Käppeli (J. Comput. Phys. 336 (2017), 104–127). This DG-like method ensures the globally divergence-free evolution of the magnetic field.
The method is built around three building blocks. The first building block consists of a divergence-free reconstruction of the magnetic field. The second building block consists of a DG-like formulation of Faraday’s law that provides a weak-form interpretation of Stokes’ law (as opposed to traditional DG, which relies on Gauss’s law). To provide a physically consistent electric field for the update of Faraday’s law, we use the third building block, which consists of a multidimensional Riemann solver that is evaluated at the edges of the mesh. We recognize that the limiting of facial variables makes the design of the MHD limiter very different from the usual DG limiter. As a result, a limiter strategy is presented for DG schemes which retains the traditional DG limiting approach while building into it a positivity-enforcement step and a step that updates the facial modes in a constraint-preserving fashion. This limiter is crucial to the robust and physically consistent operation of our DG scheme for MHD even at high orders.
It is shown that our schemes meet their design accuracies at second, third, and fourth orders on smooth test problems. Several stringent test problems with complex flow features are presented, which are robustly handled by our DG method.
中文翻译:
全局无散度DG方案可实现理想的可压缩MHD
MHD方程的高精度解决方案在物理学,数学和工程学的各个领域都引起了极大的兴趣。高阶DG方案具有低耗散和色散以及对复杂几何形状进行建模的能力,这在各种应用中都是非常需要的。由于PDE系统具有对合约束,因此MHD方程的数值解变得具有挑战性。因此,我们为可压缩MHD构建了高阶,全局无散度的DG方案。流体变量的模式在网格区域并置;磁场分量及其高阶模位于网格的表面。流体方程式是使用经典DG演化的,而磁场是使用新颖的类似DG的方法演化的,该方法首先由Balsara和Käppeli(J. 计算机 物理 336(2017),104–127)。这种类似DG的方法可确保磁场在全球范围内无扩散。
该方法围绕三个构建块构建。第一个组成部分包括磁场的无散度重建。第二个组成部分包括法拉第定律的类似DG的表述,它提供了斯托克斯定律的弱形式解释(与传统的DG依赖高斯定律相对)。为了为法拉第定律的更新提供物理上一致的电场,我们使用了第三个构件,该构件由多维Riemann求解器组成,该求解器在网格的边缘进行评估。我们认识到,面部变量的限制使MHD限制器的设计与常规DG限制器有很大不同。结果是,提出了一种针对DG方案的限制器策略,该策略保留了传统的DG限制方法,同时在其中建立了积极性增强步骤和以保持约束的方式更新面部模式的步骤。该限制器对于我们的MHD DG方案即使在高订单量下也能保持坚固且物理上一致的运行至关重要。
结果表明,我们的方案在平滑测试问题上达到了二,三,四阶的设计精度。提出了一些具有复杂流量特征的严格测试问题,这些问题可以通过我们的DG方法得到可靠解决。