The number of the diffusion

One of the number of no dimensions that are important in discussing the numerical stability on the occasion of the numerical analysis of the diffusion equation using the positive elucidation with the number of the diffusion (I do not cover it I breathe British: diffusion number). Number of the diffusion d is defined in the next expression.

${\displaystyle d=k{\dfrac {\Delta t}{(\Delta x)^{2}}}}$

Here

• k ：Diffusivity
• Δt ：Time distance in the analysis
• Δx ：Interval of the space direction

である.

Table of contents

Derivation

A one-dimensional diffusion equation:

${\displaystyle {\dfrac {\partial {u}}{\partial t}}=k{\dfrac {\partial ^{2}{u}}{\partial {x}^{2}}}}$ 

Here

• t ：Time
• x ：Coordinates in space

I think about を. I am as follows when I make a diffusion equation disintegration using law for a difference.

${\displaystyle {\dfrac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}=k{\dfrac {u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}}{(\Delta x)^{2}}}}$ 

When rewrite this expression using number of the diffusion d; quantity of physics u_i^{n +1} in time step n +1

${\displaystyle u_{i}^{n+1}=u_{i}^{n}+d(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n})}$ 

I can express と.

Evaluation of the stability by the number of the diffusion

When I calculate a diffusion equation using law for positive elucidation particularly a difference, I can evaluate numerical stability of the analysis by stability analysis of von Neumann by size of the number of the diffusion. To push forward analysis for stability

${\displaystyle d\leq {\frac {1}{2}}}$ 

There is である need. This expression is renewed as follows.

${\displaystyle \Delta t\leq {\frac {1}{2}}{\dfrac {(\Delta x)^{2}}{k}}}$ 

In other words, it is stable and cannot analyze it if I do not lower space Δt than the value that there is at time. When I make space resolution Δx small that precision is good and performs analysis, Δt must lower it by the square, and this condition becomes very severe.

References

• Norio Takeuchi, Kazuo Kashiyama, Kenjiro Terada "calculation dynamics" Morikita Shuppan, September, 2003. ISBN 4-627-91801-1。
• Kozo Fujii "numerical value numeration Tokyo University publication society of the hydrodynamics", April, 1994. ISBN 9784130628020。

Allied item

• Diffusion equation
• The number of the dziggetai

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