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The following two equations represent a simple model of a plasma where ions are immobile.

$$n\frac{\partial u}{\partial t}+nu\frac{\partial u}{\partial x}=n\frac{d\phi}{dx}-\theta\frac{\partial n}{\partial x}$$

$$\frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nu)=0$$

Here, $\theta$ is a constant denoting the temperature of the electron fluid and $\phi$ is the electric potential function obtained from an elliptic PDE (Poisson equation).

I'm looking for a (preferably) simple finite difference scheme to solve this model numerically. In particular, I'd like to see plasma oscillations.

I recently became familiar with Lax–Friedrichs method and other explicit methods for solving hyperbolic PDEs. As far as I know they can be used to solve a system of the form

$$\frac{\partial v}{\partial t}+\frac{\partial f(v)}{\partial x}=0$$

where in general $v$ and $f$ are both vector-valued functions. I'm not sure if the equations in question can be written in this form.

Let's consider two special cases. If $\theta=0$ and $n\neq 0$, the first equation can be written as

$$\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}\left(\frac{1}{2}u^2-\phi\right)=0$$

And if we neglect the electric field term, the first equation can be written as

$$\frac{\partial}{\partial t}(nu)+\frac{\partial}{\partial x}\left(nuu+\theta n\right)=0$$

Both of these equations seem to be in the form mentioned above.

What is the standard or most common finite difference scheme to solve the set of the two PDEs at the beginning of this question?

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