American Institute of Mathematical Sciences

March  2012, 5(1): 155-184. doi: 10.3934/krm.2012.5.155

Second order all speed method for the isentropic Euler equations

 1 Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, No. 800 Dong Chuan Road, Minhang, Shanghai 200240, China

Received  September 2011 Revised  September 2011 Published  January 2012

Standard hyperbolic solvers for the compressible Euler equations cause increasing approximation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5].
The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semi-discrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the momentum equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes explicit when the time step is small enough.
Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly.
Citation: Min Tang. Second order all speed method for the isentropic Euler equations. Kinetic & Related Models, 2012, 5 (1) : 155-184. doi: 10.3934/krm.2012.5.155
References:

show all references

References:
 [1] Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 [2] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [3] Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583 [4] Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056 [5] Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1 [6] Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051 [7] Yanmin Mu. Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2014, 7 (4) : 739-753. doi: 10.3934/krm.2014.7.739 [8] Fucai Li, Yanmin Mu. Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1669-1705. doi: 10.3934/dcds.2018069 [9] Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122 [10] Andrea Natale, François-Xavier Vialard. Embedding Camassa-Holm equations in incompressible Euler. Journal of Geometric Mechanics, 2019, 11 (2) : 205-223. doi: 10.3934/jgm.2019011 [11] Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 [12] María J. Martín, Jukka Tuomela. 2D incompressible Euler equations: New explicit solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4547-4563. doi: 10.3934/dcds.2019187 [13] Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387 [14] Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030 [15] Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068 [16] Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51 [17] Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021 [18] Yachun Li, Shengguo Zhu. On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3059-3086. doi: 10.3934/dcds.2015.35.3059 [19] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [20] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17

2018 Impact Factor: 1.38