Hyperbolic Equation Instability

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SUMMARY

The discussion focuses on addressing numerical instability in the calculation of the derivative of a distribution function, specifically in the context of an MHD accelerator simulation. The instability arises from using central difference methods in regions with significant discontinuities in the electric field. Participants recommend utilizing MUSCL and TVD schemes, as well as implementing a hybrid approach that combines central differencing with low-order upwinding schemes to enhance stability near discontinuities.

PREREQUISITES
  • Understanding of numerical methods for partial differential equations
  • Familiarity with central difference and upwinding schemes
  • Knowledge of MUSCL (Monotonic Upstream-centered Schemes for Conservation Laws)
  • Basic principles of magnetohydrodynamics (MHD)
NEXT STEPS
  • Research the implementation of MUSCL schemes in computational fluid dynamics (CFD)
  • Study TVD (Total Variation Diminishing) schemes for shock capturing
  • Explore hybrid numerical methods combining central differencing and upwinding techniques
  • Investigate local gradient methods for adaptive numerical scheme switching
USEFUL FOR

Researchers and engineers working in computational fluid dynamics, particularly those involved in simulations of magnetohydrodynamic systems and numerical methods for solving partial differential equations.

Tempa
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Hello,

I'm trying to calculate the following equation which is the derivative in 'x' of a distribution function:
d(dxF)/dt = d(Efield.(dvxF))/dx

The problem comes because the right hand of the equation is solved by using central difference, but there is a zone where there is a discontinuity in the electric field. The electric field is not exactly the same on each point (but more or less the same order of magnitude) but there is a zone where I have in increment so the difference between two adjacent grid is so big that creates an instability. Is there any way to smooth the numerical scheme?

= (E[i+1][j]. dvxF[i+1][j]-E[i-1][j].dvxF[i-1][j])/dx

This is more or less how it goes. Imagine there is a point where the difference between E[i+1][j] and E[i-1][j] is so big that, creating a big gradient in the dxF value.

I've been studying the MUSCL and TVD schemes but i don't quite understand well the procedure
 
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Did you really mean to put this in the aerospace engineering forum?
 
Yes because I'm simulating an MHD accelerator, is just a specific problem I have. But if its not the correct place I can post it in another forum. Sorry about that
 
No I was just curious. It is just not something I am familiar with I suppose. Then again, I am not a numerical methods guy.
 
TVD schemes are a good place to start. If you search on CFD "Shock Capturing" Schemes, you'll find a lot of resources that should be applicable.

Essentially, at the core, what you would like to do is use local gradients to switch between your central differencing scheme (no numerical dissipation) and a low-order upwinding scheme (lots of numerical dissipation).

As a first cut, try solving using both methods and use a weighted average of a local gradient to determine the weightings. At that point near your discontinuity, force your solver to use the low-order scheme; this could help your stability.
 
minger, thank you very much for your advise
 

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