守恒流体力学方程组的熵不等式

The Entropy Inequality of the Conservative Gas Dynamic Systems

关于双曲型守恒方程组ut+fx=0,若存在标量熵对(U,F)满足熵不等式 Ut+Fx≤0,则原方程存在弱解.本文目的在于将此结果应用到流体力学方程组.文中先求其半离散差分逼近式及离散熵对[Ut,F0j +G1j ],通过引入适当的调节参数β及调节流函数f1j 的计算方法,使取极限后熵不等式恒为满足.因而使差分逼近式的解收敛于弱解,并举出一些数值算例说明本方法的有效性.

With respect to the conservative hyperbolic systems u_t+f_x=0,if there exists entropy couple(U,F) which satisfies the entropy inequality U_t+F_x≤0,the hyperbolic system possesses a weak solution.After applying this result to the conservative gas dynamic systems,the semi-discrete finite difference approximation and the discrete entropy couple[U_t,F^0_j+G^1_j] is searched.Then the crucial numerical method of adequate adjusting parameter β and the adjusting discrete flux f^1_j are introduced such that the entropy inequality is always satisfied and solution of the finite difference approximation approaches to certain weak solution as h→0.Some numerical examples are used to demonstrate the efficiency of the propose method.