ODE solver with WENO scheme (weighted essential non-oscillatory)

Hey everyone,

this is rather a very specific question:

I want to solve a bunch of DAE in Matlab using ode-solver (ode15s in my case for stiff problems).

The system depends in time t and location z.

For that, I had to transform the PDAE into an DAE by discretizing in z-direction.

First I used the upwind scheme for disretization, but had the problem of artifical oscillations. So I tried to implement the WENO scheme, which is described here: meatballbw.ps (nd.edu)

After implemeting the WENO scheme, I even faced more oscillation.

Is anyone familiar with the WENO scheme or has another approach, how to solve artifical oscillation with ODE solvers. I also tried to implement a non-constant Jacobi matrix but it failed: ODE15s with non-constant Jacobian - (mathworks.com) .

Thank you for your help.

[Torsten]

What are the equation(s) you are trying to solve ?

What is your code ?

Upwind schemes don't produce artificial oscillations - maybe you mean artificial diffusion for 1st order upwind schemes.

[Berry]

Dear @Torsten, thank you for your help.

Actually I solved this oscillation problem with the jacobian implementation in our last discussion: https://de.mathworks.com/matlabcentral/answers/1772755-ode15s-with-non-constant-jacobian

Just one question to understand the problem better. In the following paper they say:

I proplaby missunderstood the red part. I thought, that using first order upwind with high discretization numbers is a high order scheme, which can cause artifical ocsillation...

[Torsten]

I thought, that using first order upwind with high discretization numbers is a high order scheme, which can cause artifical ocsillation...

No. Also if you use many grid points (I think this is what you mean by "high discretization numbers"), a first-order scheme remains a first-order scheme. It will not cause artificial oscillations.

If you don't choose enough grid points, a first-order scheme tends to smear out sharp gradients. This is meant by the technical term that they "cause numerical dispersion".

[Berry]

Perfect. Thanks for the clarification.