# Hyperbolic Equation Instability

• Tempa
In summary: I will give it a try and see if that works. In summary, the author is trying to solve an equation for the derivative of a distribution function, but there is a zone where the electric field is not exactly the same. He suggests using a low-order upwinding scheme near the discontinuity to help stability.
Tempa
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

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

## 1. What is a hyperbolic equation instability?

A hyperbolic equation instability is a type of instability that occurs in mathematical models described by hyperbolic partial differential equations. These equations are used to describe systems that involve waves or oscillations, such as fluid dynamics, electromagnetism, and quantum mechanics.

## 2. How does a hyperbolic equation instability manifest in a system?

A hyperbolic equation instability manifests as a rapid growth or amplification of small fluctuations in a system. This can lead to chaotic or unpredictable behavior, making it difficult to accurately model and predict the behavior of the system.

## 3. What are the causes of hyperbolic equation instabilities?

There are several factors that can contribute to hyperbolic equation instabilities, including nonlinear interactions, boundary conditions, and initial conditions. These instabilities can also be caused by errors in the model or numerical methods used to solve the equations.

## 4. How can hyperbolic equation instabilities be controlled or mitigated?

There are various techniques that can be used to control or mitigate hyperbolic equation instabilities. These include adjusting boundary conditions, using more accurate numerical methods, and incorporating stabilizing terms into the equations. Additionally, careful validation and verification of the model can help identify and prevent potential instabilities.

## 5. Are there any real-world applications of hyperbolic equation instabilities?

Yes, hyperbolic equation instabilities can have significant impacts in many real-world applications, such as weather forecasting, aerodynamics, and structural engineering. Understanding and controlling these instabilities is crucial for accurately predicting and designing systems in these fields.

Replies
3
Views
2K
Replies
2
Views
978
Replies
7
Views
2K
Replies
16
Views
426
Replies
3
Views
1K
Replies
1
Views
1K
Replies
13
Views
1K
Replies
0
Views
575
Replies
5
Views
2K
Replies
30
Views
1K