Quantum (fluid) Equation With The Debye Length Limit The Asymptotic Format

In this paper we deal with the zero Debye length asymptotic of solutions of isentropic quantum hydrodynamic equations for semiconductors. In this limit, explicitnumerical schemes suffer from severe numerical constraints because of the small Debye length. In our work, we propose an implicit scheme for the quantum hydrodynamic model in this limit. This scheme overcomes the constraints of time step, and it has the same numerical cost as the explicit scheme. At the end of this paper, we also present some numerical tests. In section one, the model of zero Debye length asymptotic of isentropic quantum hydrodynamic equationsand the reformulated Possion equationare deduced. Then, the Euler-Possion systems are changed intoand we consider the discretiation of time and space of the system above.In section two, the main aim is the asymptotic preserving scheme of the sys- tem(0.3) as followswhere, q =ρuBecause the scheme is related with the number of three time steps , we have to use the classical scheme at the first and the second steps, the classical scheme isThe equation(0.5) appears as an elliptic problem which allows to compute provided that all quantities up to time step n are known. Then the second equation of (0.4) allows to compute qⁿ⁺¹, at last, using the first equation of (0.4) to compute the value ofρat time step n + 1, i.e.ρⁿ⁺¹.In addition, the scheme is not fully implicit in the treatment of the electric field source term since the termρ▽Φis approximated byρⁿ▽Φⁿ⁺¹ at the left of the second equation of (0.4). A fully implicit treatment e.g.ρⁿ⁺¹▽Φⁿ⁺¹ would make the computation of qⁿ⁺¹ to acquire the density at step n + 1, it is a bad trend. Actually, a full implicit scheme is not necessary.(see[2]) At last, we point out that the relaxation-time term is treated by the average of two time steps.In the last section, we give some numerical tests. We consider the stationary solution of the zero debye length asymptotic of the quantum hydrodynamic system which is given by W⁰ = (ρ⁰ = 1,q⁰ = 1,Φ⁰ = 0). The most important test-case is the perturbation of this steady state. We present a simulation on the domain (0,1) with periodic boundary conditions for the Euler systems and with homogeneous Dirichlet boundary conditions for the Poisson equation. We consider a perturbation of W⁰, which is given by the following:whereδ= 10^-2 is the perturbation amplitude. We select the parameters as follows:γ= 5/3,ε= 0.01, andλ= 0.01, that is to say,ω= 10³. We will see that when△t≥ω^-1, the classical scheme is not stable, however, the asymptotic scheme does well.