Unconditionally Stable Implicit Difference Scheme for the Hydrodynamical Equations.

摘要

We solve two hydrodynamical problems. The first involves a shock wave, a contact discontinuity, and a rarefaction wave using an unconditionally stable finite difference scheme. The Courant condition is satisfied everywhere except in one zone behind the shock, where it is violated by factors of 10 and 100. The nonlinear difference equations are solved by Newton's method. The total number of Newton iterations to get to a certain time is apparently independent of the degree to which the normal stability condition is violated in the one zone. The second problem involves two rarefaction waves moving in opposite directions. One wave moves in a region where the Courant condition is violated by a factor of approximately two. The other wave moves in a region where the Courant condition is satisfied. Numerical results are compared with the analytical solution. An examination of several runs indicates one explicit time step is about five times as fast as one implicit time step. Therefore, use of the implicit method is indicated when the Courant condition is violated by a factor of 5 or more.