A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential (Communications in Computational Physics, 2020)
The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep thisbalance exactly due to their unbalanced truncation errors. In this study, we introducean auxiliary variable which becomes constant at isothermal hydrostatic equilibria andpropose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Throughreformulating the convection term and the force term via the auxiliary variable, zeronumerical 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 newscheme. The results confirm that, the new scheme can preserve the exact hydrostaticsolution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process ofconverging towards hydrostatic equilibria from a highly unbalanced initial condition.The ultimate state of zero velocity and constant temperature is achieved up to machineaccuracy. As demonstrated by the numerical experiments, the current scheme is verysuitable for small amplitude perturbation and long time running under gravitationalpotential.
Keywords: Well-balanced; source term; gravity; gas kinetic scheme.