Application of the method of lines to unsteady compressible Euler equations

In the sense of method of lines, numerical solution of the unsteady compressible Euler equations in 1D, 2D and 3D is split into three steps: First, space discretization is performed by the first-order finite volume method using several approximate Riemann solvers. Second, smoothness and Lipschitz continuity of RHS of the arising system of ordinary dimensional equations (ODEs) is analysed and its solvability is discussed. Finally, the system of ODEs is integrated in time by means of implicit and explicit higherorder adaptive schemes offered by ODE packages ODEPACK and DDASPK, by a backward Euler scheme based on the linearization of the RHS and by higher-order explicit Runge-Kutta methods. Time integrators are compared from several points of view, their applicability to various types of problems is discussed, and 1D, 2D and 3D numerical examples are presented.