A representation and approximation of the solutions of hyperbolic differential equations
A system of quasilinear hyperbolic partial differential equations is discussed for a qualitative study of weak solutions and for developing a suitable numerical scheme. The Stieltjes transform and its properties are used to reduce the system of equations to a fixed point equation which is put in a computationally feasible form after comparing it with the method of characteristics. A discontinuous or piecewise continuous profile may be assumed for the initial approximation. The equations of fluid flow in an ideal shock tube are discussed using this method and the following improvements are noted: Spurious oscillations are absent, the resolution and accuracy of the location of the discontinuities are excellent; the constant state is adequately realized; and the diffusion is minimal.
scientific article; zbMATH DE number 4045071