Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics
Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics
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发表时间：
1959
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通讯作者：
S. Godunov;I. Bohachevsky
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作者：
S. Godunov;I. Bohachevsky
用于流体动力学方程数值计算的特征线法具有很大的非标准性，因此不适合在电子计算机上自动计算，特别是对于含有大量激波和接触间断的问题。1950年，诺伊曼和里希特迈尔提出使用差分方程求解流体动力学方程，其中人为地引入了粘性;这具有在几个网格点上涂抹激波的效果。然后，建议以普通方式进行跨冲击波的计算。1954年，拉克斯发表了适用于计算激波的“三角形”格式。这个方案的一个缺点是它不允许用任意精细的时间步长进行计算（与声速除以空间步长相比），因为它会将任何初始数据转换为线性函数。此外，该方案消除了接触不连续性。本文的目的是选择一个计划，这是在某种意义上最好的，它仍然允许计算整个冲击波。这种选择是线性方程组，然后类推的计划适用于流体动力学的一般方程。按照这个方案，我们在苏联的电子计算机上进行了大量的计算。为了检验，将其中一些计算与特征线法进行的计算进行了比较。结果的一致性完全令人满意。
The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.