Solving two coupled 1D advection equations - stability problem

Questions
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1

I am encoutering some difficulties when I try to solve a system of two coupled advection equations in 1D:

Screenshotfrom2022-08-2315-54-02.png
Screenshotfrom2022-08-2315-54-02.png358×118 13.3 KB

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where na and ni are the searched functions; v_a, v_i and beta are constant coefficients.

Firstly, I try solve the simpler system of two non-coupled transport equations, by simpling putting the right hand sides of the equations to 0. For simplicity, I set the initial conditions to be sin(x) and left boundary conditions to be 0. Thus one obtains a pretty straightforward system of two uncoupled transport equations. The solutions obtained for this system seem reasonable:

Open in Wolfram Cloud ↗

Then, I introduce some exponential growth along the characteristics. At this point, the solution looses stability and some oscillations are seen in the solutions:

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The next step would be to introduce some coupling between the equation. Any suggestions how to prevent the oscillations from arising?

2

Hi Olgierd,

The step size needs to be reduced, not set to Infinity. Try MaxStepSize -> 0.001.

3

Dear Rohit,

Thank you for your response. Yes, this was an error - correcting it prevents the oscillations from arising, as shown in the attached notebook:

Open in Wolfram Cloud ↗

However, going one step further and adding the coupling on the right-hand side of the equations:

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the solution diverges. Any suggestions on how to make the solution converge?
[1]: Wolfram Cloud Document
[2]: Wolfram Cloud Document

4

I have double-checked the numerical solution obtained by Mathematica with analytical solutions for simple boundary conditions (e.g. such as x(2-x)). Looks like Mathematica is calculating everything correctly - the analytical solutions suggest that singularities are indeed present in the points stated in the console.

Closing the thread.