Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis
Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis
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DOI：
10.1007/s000330050150
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发表时间：
1999-03
期刊：
影响因子：
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通讯作者：
Tao Tang;Kun Xu
中科院分区：
文献类型：
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作者：
Tao Tang;Kun Xu
基于碰撞BGK模型的数值格式是近年来发展起来的。本文研究Euler方程的一阶BGK格式。特别注意的是找到CFL的条件下，该计划是积极的保持（即密度和内部能量保持非负）。一阶BGK格式是无碰撞（即动力学通量分裂格式）和碰撞方法的线性组合。我们表明，无碰撞的方法保持了标准CFL条件下的密度和内能的正性。虽然无碰撞方法具有保正性，但由于相应的方案是基于两个半麦克斯韦方程，因此它引入了大的内禀耗散和热传导。为了减少粘性误差，一个明显的方法是使用精确的麦克斯韦，这导致碰撞玻尔兹曼格式。CFL的条件也被发现的碰撞的方法，这在文献中提供的测试问题，效果很好。然而，通过考虑一个反例，我们发现碰撞方法并不总是保正的。BGK型格式综合了两种方法的优点，即耗散较小的（碰撞）格式和耗散较大但保正的（无碰撞）格式。
Numerical schemes based on the collisional BGK model have been developed in recent years. In this paper, we investigate the first-order BGK schemes for the Euler equations. Particular attention is given to finding CFL-like conditions under which the schemes are positivity-preserving (i.e. density and internal energy remain nonnegative). The first-order BGK schemes are linear combinations of collisionless (i.e. kinetic flux-splitting scheme) and collisional approach. We show that the collisionless approach preserves the positivity of density and internal energy under the standard CFL condition. Although the collisionless approach has the positivity-preserving property, it introduces large intrinsic dissipation and heat conductions since the corresponding scheme is based on two half Maxwellians. In order to reduce the viscous error, one obvious method is to use an exact Maxwellian, which leads to the collisional Boltzmann scheme. An CFL-like condition is also found for the collisional approach, which works well for the test problems available in literature. However, by considering a counterexample we find that the collisional approach is not always positivity-preserving. The BGK type schemes are formed by taking the advantages of both approaches, i.e. the less dissipative scheme (collisional) and the more dissipative but positivity-preserving scheme (collisionless).