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 plotOutput of simultation where $\mathrm{\color{blue}{blue}}$, $\mathrm{\color{red}{red}}$ and $\mathrm{\color{green}{green}}$ lines correspond grids with $(1024,\,512,\,256)$ poins respectively.


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?


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


1 Answer 1


Your numerical solution is probably just getting more accurate as you increase the number of grid points. Do you know or have you tried to derive the analytic (exact) solution for this problem? By looking at your plots, it seems like the exact solution has a shock (discontinuity) occurring at around x = 4.5, and the numerical method is resolving it with greater accuracy as you increase the number of grid points. The numerical solution will always depend on the grid you use.


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