Von Neumann Analysis: Refresh Numerical Science | Any Help Appreciated

In summary, the conversation discusses the use of Von Neumann analysis to solve a numerical equation involving convection. The goal is to find the smallest possible diffusion while still maintaining stability. Based on some old notes, it is suggested to simplify the equation using Taylor series and manipulating the j+1 and j terms. However, the last step where the diffraction term is finalized remains unclear and the speaker is seeking some guidance on how to proceed.
  • #1
liquidFuzz
97
3
I'm trying to refresh some numerical science stuff. Von Neumann analysis, if I take a slimmed down equation, convection. [tex] \frac{∂u}{∂t}+a ∇ u =0[/tex] If I'm using Euler forward, [tex] \frac{u^{n+1}-u^n}{\Delta t}+\frac{a}{2h} \left( u_{j+1}^n -u_{j-1}^n \right) =0[/tex] For [itex]\hat{u}^n = G^n\hat{u}^0[/itex] a growth factor [itex]|G|\leq1[/itex] is sufficient. This gives, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \beta h \left( \frac{u_{j+1}^n-2u_j^n + u_{j-1}^n}{h^2} \right)[/tex] With the smallest possible diffusion but still stable, [itex]\beta \geq \frac{a}{2}[/itex] becomes [itex]\beta = \frac{a}{2}[/itex]. If this is plugged into the equation, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \frac{a}{2h} \left( {u_{j+1}^n-2u_j^n + u_{j-1}^n} \right)[/tex] Now, according to some old notes I have the result is, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \frac{1}{h} \left( {au_j^n -a u_{j-1}^n} \right)[/tex] Well this is somewhat vague for me. Specially the last step where the defraction term is finalised.

Any pointers of help would be appreciated, specially that last step.
 
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  • #2
I faintly remember that sometimes you could simplify things using a Taylor series going forward in j and a taylor series going backwards in j, then subtracting (or adding) one from the other. At least something along those lines. Considering the j-1 term remained, I'd consider doing the taylor series over the j+1 term (eq1) and the j term (eq2) and then multiplying eq2 by a factor of 2 and then adding (or subtracting) eq1 from eq2 and see what happens. You can probably ignore higher order terms which are negligible (or your only looking at a first order scheme).
 
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1. What is Von Neumann Analysis?

Von Neumann Analysis is a mathematical method used to analyze the stability and accuracy of numerical methods. It was developed by renowned mathematician John von Neumann in the 1940s and is commonly used in the field of numerical science.

2. Why is Von Neumann Analysis important in numerical science?

Von Neumann Analysis allows scientists to predict the behavior of numerical methods, such as algorithms or computer programs, before they are implemented. This helps in identifying potential errors and improving the accuracy and efficiency of these methods.

3. What are the main components of Von Neumann Analysis?

The main components of Von Neumann Analysis are the stability region, the amplification factor, and the phase shift. The stability region is a graphical representation of the numerical method's stability, while the amplification factor and phase shift help in determining the accuracy and convergence of the method.

4. How is Von Neumann Analysis performed?

Von Neumann Analysis involves using mathematical equations and principles, such as the Fourier transform, to analyze the behavior of a numerical method. This analysis often involves creating a stability diagram and performing calculations to determine the method's stability and accuracy.

5. What are the limitations of Von Neumann Analysis?

Von Neumann Analysis is limited to linear problems and may not accurately predict the behavior of non-linear systems. Additionally, it assumes that the numerical method is applied to a continuous system, which may not always be the case in real-world scenarios.

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