Attenuation numerical instability via numerical diss. nonlinear ODE

[nickthequick]

Hi,

I'm solving a problem numerically that takes the form

$Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0$,

where $(Q_{ij},S_{ijk},V_i)$ are all functions of the dependent variables $y_i$. The dependent variables are all functions of the variable t. The resolution of this spectral model is controlled by the number of $y_i$, denoted N.

Now, when these equations become stiff, or I integrate for long times, numerical instability becomes apparent. If I increase the resolution, these spurious growths are attenuated, but it is not feasible to make N large enough so that they don't appear at all.

If one looks at the time history of the higher modes, it seems like their large growth corresponds to the onset of the instability, so I want to attenuate these (un-physical) fast growing components of the solutions.

To this end, I have added a term to the leading order term, so that the equation takes the form

$Q_{ij}( \ddot{y}_j +\nu j^2 \dot{y}_j)+S_{ijk}\dot{y}_j\dot{y}_k +V_i=0$,

where $\nu$ is a numerical viscosity to be prescribed later on. This does an OK job at attenuating the instability (e.g. I get accurate solutions for an twice as long), based on how I choose $\nu$, and the constraints of the problem. I have played around with the power of j as well, and found varying success in different contexts.

The issue is, for long times this scheme breaks down, and the solution greatly deviates from the expected solution, making me think I'm not doing addressing the issue very rationally. Indeed, I am doing all of this very naively, and am having trouble finding similar types of problems in the literature.

Does anyone have experience damping numerical instabilities in equations that take this form? In particular, is there a way to essentially add a dissipation that removes these fast growing modes? References to relevant literature are also greatly appreciated.

Thanks!

Nick

 

[eljose79]

Hi Nick,

I have encountered similar issues with numerical instabilities in stiff equations before. One approach that has worked for me is to use a higher-order numerical scheme, such as a Runge-Kutta method, which can better handle stiff equations. Another approach is to use a variable time step, where the step size is adjusted based on the stability of the solution. This can help in cases where the stiffness is not uniform throughout the solution.

In terms of adding dissipation to remove fast growing modes, one method that has been successful in the past is to use artificial viscosity. This involves adding a term to the equations that acts as a source of dissipation, effectively smoothing out the solution and removing any unstable modes. The challenge here is to choose the right amount of artificial viscosity, as too little will not have an effect and too much will overly dampen the solution. There is a lot of literature on this topic, so I would recommend doing some research and perhaps consulting with colleagues or experts in the field to find the best approach for your specific problem.

I hope this helps and good luck with your numerical simulations!