# Grid dependence of a numerical model

## Statement of the problem

Suppose, we consider the following model $$\begin{array}{l} (1)~\mathbf{u}_t + \mathbf{F(u)}_x = \mathbf{S}(\mathbf{u},\mathbf{w}), \\ (2)~\mathbf{w}_x = \mathbf{P}(\mathbf{w},\mathbf{u}). \end{array}$$ Computational domain to be as follows $$(x,t)=[0,L]\times[0,T]$$. Let's consider Neumann boundary conditions (BC) for (1) and initial value problem (IVP) for (2).

Algorithm of simulation as follows: on each time step $$t^n$$ solve (2) at first and then solve (1).

One could assume that $$\mathbf{u} = [n_d\,v_d]^T$$ and $$\mathbf{w} = [n_e\,E]^T$$ and obtaion the corresponding plot where $$\mathrm{\color{blue}{blue}}$$, $$\mathrm{\color{red}{red}}$$ and $$\mathrm{\color{green}{green}}$$ lines correspond grids with $$(1024,\,512,\,256)$$ poins respectively.

## Question

One could realize that the shape of the solution depends on the number of grid points, i.e. grid depence achieved. How to avoid it?

## P.S.

Eq. (1) was solved by smoothed Lax-Wendroff scheme (LxW) and eq. (2) was solved by Runge-Kutta scheme of 4th order.