Hyperbolic Equation Instability

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Discussion Overview

The discussion revolves around the numerical instability encountered when solving a hyperbolic equation related to a distribution function in the context of magnetohydrodynamics (MHD). Participants explore methods to smooth the numerical scheme to address issues arising from discontinuities in the electric field.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a specific equation involving the derivative of a distribution function and notes instability due to a significant gradient caused by discontinuities in the electric field.
  • Another participant questions the appropriateness of the forum for the topic, expressing unfamiliarity with the subject matter.
  • A participant suggests that Total Variation Diminishing (TVD) schemes could be beneficial and recommends exploring "Shock Capturing" schemes in computational fluid dynamics (CFD) to manage stability issues.
  • A proposed approach involves using local gradients to switch between central differencing and a low-order upwinding scheme to enhance stability near discontinuities.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to address the instability, and multiple strategies are discussed without resolution.

Contextual Notes

The discussion highlights the challenges of numerical methods in the presence of discontinuities and the need for further exploration of specific schemes like MUSCL and TVD, which may have varying applicability depending on the problem context.

Who May Find This Useful

Individuals interested in numerical methods for solving hyperbolic equations, particularly in the fields of aerospace engineering and magnetohydrodynamics, may find this discussion relevant.

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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