Volume 12, Issue 4
An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

Jeffrey Haack, Shi Jin & Jian-Guo Liu

Commun. Comput. Phys., 12 (2012), pp. 955-980.

Published online: 2012-12

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  • Abstract

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

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@Article{CiCP-12-955, author = {}, title = {An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations}, journal = {Communications in Computational Physics}, year = {2012}, volume = {12}, number = {4}, pages = {955--980}, abstract = {

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.250910.131011a}, url = {http://global-sci.org/intro/article_detail/cicp/7320.html} }
TY - JOUR T1 - An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations JO - Communications in Computational Physics VL - 4 SP - 955 EP - 980 PY - 2012 DA - 2012/12 SN - 12 DO - http://dor.org/10.4208/cicp.250910.131011a UR - https://global-sci.org/intro/article_detail/cicp/7320.html KW - AB -

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

Jeffrey Haack, Shi Jin & Jian-Guo Liu. (2020). An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations. Communications in Computational Physics. 12 (4). 955-980. doi:10.4208/cicp.250910.131011a
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