Well-Balanced Methods for Compressible Euler Equations with Gravitational Force that Preserve Transonic Stationary Solutions

In some previous works, the authors introduced a general methodology to design high-order well-balanced finite-volume methods for one-dimensional systems of balance laws based on the use of well-balanced state reconstructions. Local steady states have to be computed in the stencil of every cell, which is done by solving the ODE system satisfied by stationary solutions using collocation Runge-Kutta (RK) methods. A strategy to deal with resonant problems that is general, albeit problem-dependent, was also proposed and applied to the shallow-water model: the derivative of a transonic stationary solution at the sonic point can be analytically computed and this value is used in the computation of local steady state when a sonic point is detected. In this work, this technique is applied to the compressible Euler equations with gravitational force and the numerical results are checked.