Volume 28, Issue 3
A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

Songze Chen, Zhaoli Guo & Kun Xu

Commun. Comput. Phys., 28 (2020), pp. 902-926.

Published online: 2020-07

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  • Abstract

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

  • Keywords

Well-balanced, source term, gravity, gas kinetic scheme.

  • AMS Subject Headings

76M99, 86A10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-902, author = {Chen , Songze and Guo , Zhaoli and Xu , Kun}, title = {A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {3}, pages = {902--926}, abstract = {

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0067}, url = {http://global-sci.org/intro/article_detail/cicp/17661.html} }
TY - JOUR T1 - A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential AU - Chen , Songze AU - Guo , Zhaoli AU - Xu , Kun JO - Communications in Computational Physics VL - 3 SP - 902 EP - 926 PY - 2020 DA - 2020/07 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0067 UR - https://global-sci.org/intro/article_detail/cicp/17661.html KW - Well-balanced, source term, gravity, gas kinetic scheme. AB -

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

Songze Chen, Zhaoli Guo & Kun Xu. (2020). A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential. Communications in Computational Physics. 28 (3). 902-926. doi:10.4208/cicp.OA-2019-0067
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