Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for a nonlinear Euler-Poisson system (Q1585011)

The paper deals with the Euler-Poisson system \[ \begin{aligned} \partial_t n+\partial_x(nu)&=0,\\ \partial_t(nu)+\partial_x(nu^2+p(n))&=-\sigma n(u-u_*)-n\partial_x\varphi,\\ -\partial_{xx}^2\varphi& = n - f(\varphi)+b. \end{aligned} \] where \(p\) obeys a gamma law with \(\gamma>1\), \(f\) is an increasing function of the electric potential \(\varphi\) (vanishing at \(0\) and unbounded at \(\infty\)), \(\sigma\) and \(u_*\) are given bounded functions of \(x\) with \(\sigma\geq 0\), and \(b\) is a bounded integrable function of \(x\) on the whole real line. This system is endowed with bounded initial data \(n_0\), \(u_0\) with \(n_0\) being nonnegative and compactly supported, together with homogeneous boundary condition for \(\varphi\) at \(\infty\). The main result is the global existence of weak entropy solutions. A similar result was previously shown by the author and \textit{S. Cordier} [RAIRO, Model. Math. Anal. Numer. 32, No. 1, 1-23 (1998; Zbl 0935.35119)] in the isothermal case (\(\gamma=1\)). The proof is based on a careful analysis of the nonlinear Poisson equation and finite difference approximations of the Euler equations. Convergence is inspired from works by G. Q. Chen and co-workers on isentropic gas dynamics.