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Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for a nonlinear Euler-Poisson system. (English) Zbl 0965.65113
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 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.
Reviewer: S.Benzoni (Lyon)

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35Q60 PDEs in connection with optics and electromagnetic theory 78M20 Finite difference methods applied to problems in optics and electromagnetic theory 78A35 Motion of charged particles
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