Fluid mechanics: Explicit second order splitting up (disintegration) schemes which produce shock contours without oscillation

A general class of explicit second order schemes is applied to the solution of the Cauchy problem with shock wave for a model. The equivalent equations or equations discretized to the third order by the disintegration schemes are given. A subclass of optimum schemes, i.e., dissipative with a minimum dissipation coefficient, and a subclass of error minimizing schemes are derived. A problem in which the initial compression contour connecting two constant states is transformed after a few cycles into a shock wave which moves parallel to the initial direction is solved numerically. It is thus demonstrated that the subclasses enable oscillation free shock contours to be obtained without the addition of artificial viscosity.