Stability Theory for Difference Approximations of Euler-Korteweg Equations and Application to Thin Film Flows
Stability Theory for Difference Approximations of Euler-Korteweg Equations and Application to Thin Film Flows
复制标题
Euler-Korteweg 方程差分近似的稳定性理论及其在薄膜流中的应用
DOI：
--
复制
发表时间：
2013
影响因子：
2.9
通讯作者：
J. Vila
中科院分区：
文献类型：
--
作者：
P. Noble;J. Vila
研究了Euler-Korteweg方程的各种差分近似解的稳定性。这个演化偏微分方程组是一个受色散(三阶)项扰动的经典等熵欧拉系统。欧拉方程用经典格式(如Roe、Rusanov或Lax-Friedrichs格式)离散，而色散项用中心有限差分离散。我们首先证明了一个差分格式在Von Neumann意义下稳定需要一定的数值粘性。然后，我们考虑了差分逼近的熵稳定性。为此，我们引入了一个额外的未知数，密度函数的梯度。将Euler-Korteweg系统转化为受二阶斜对称项扰动的双曲组。在适当的Courant-Friedrichs-Levy条件下，证明了Lax-Friedrichs型格式的熵稳定性。此外，我们还提出了欧拉-Korteweg系统的空间离散化，该系统被视为Hamil系统。
We study the stability of various difference approximations of the Euler--Korteweg equations. This system of evolutionary PDEs is a classical isentropic Euler system perturbed by a dispersive (third order) term. The Euler equations are discretized with a classical scheme (e.g., Roe, Rusanov, or Lax--Friedrichs scheme), whereas the dispersive term is discretized with centered finite differences. We first prove that a certain amount of numerical viscosity is needed for a difference scheme to be stable in the Von Neumann sense. Then we consider the entropy stability of difference approximations. For that purpose, we introduce an additional unknown, the gradient of a function of the density. The Euler--Korteweg system is transformed into a hyperbolic system perturbed by a second order skew symmetric term. We prove entropy stability of Lax--Friedrichs type schemes under a suitable Courant--Friedrichs--Levy condition. In addition, we propose a spatial discretization of the Euler--Korteweg system seen as a Hamil...