Obtaining physically relevant shock-wave solutions with the aid of the Rothe method

Differential equations in conservation form, briefly called conservation equations, play an important part in physics. These hyperbolic partial differential equations occur in the study of nonlinear waves, if dissipation effects, such as viscosity, are disregarded. Thus, the Euler equations describing compressible flow mechanics consist of such conservation equations. A characteristic feature of these nonlinear equations is the occurrence of discontinuities, such as shock waves, in the solutions. Harten et al. (1976) have studied approaches for the solution of problems involving the considered equations. The present investigation is concerned with the possibility to employ for the study of these problems a method developed by Rothe (1930). Attention is given to an employment of the Rothe method in the case of the Cauchy problem for the scalar conservation equation, the applicability of the Rothe method to an initial-value, boundary-value problem, and the application of the Rothe method in the case of two test problems.