Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic energy preservation leading to a reduction in aliasing errors at high orders. Recently, linear stability issues have been pointed out for these schemes for a density wave problem and we investigate this behaviour for some standard split forms. By deriving linearized equations of split form schemes, we show that most existing schemes do not satisfy a perturbation energy equation that holds at the continuous level. A simple modification to the energy flux of some existing schemes is shown to yield a scheme that is consistent with the energy perturbation equation. Numerical tests are given using a discontinuous Galerkin method to demonstrate these results.
A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所计算物理实验室）
Simultaneous energy and entropy stability of volume-penalisation immersed-boundary methods for the compressible Navier--Stokes equations
ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空工程学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）
Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework
Southern University of Science and Technology（南方科技大学）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳数学中心）
Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
介绍了保内积范数松弛龙格-库塔法（David I. Ketcheson，\emph{Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms}，SIAM Journal on数值分析，2019）扩展到一般凸量。通过增加{ \em松弛因子}来加强关于任何凸泛函的守恒、耗散或其他解性质，该松弛参数在每一步乘以龙格库塔更新。此外，保持了其它所需的稳定性（如强稳定性保存）和效率（如低储存要求）性质。该技术可以应用于显式和隐式龙格-库塔法，只需要一个小的修改，现有的实现。在每一步的计算成本是一个额外的标量代数方程的解决方案，一个很好的初始猜测是可用的。该方法的有效性被证明分析和证明在几个数值例子中，包括应用到高阶熵守恒和熵稳定的半离散的非结构网格上的可压缩欧拉和Navier-斯托克斯方程。
