What is the difference between diffusion and dispersion?

dut_thinker · February 7, 2011, 21:49

According to my understanding, diffusion is a process caused by concentration difference. Dispersion is the combined effort of adveciton and diffusion.
Is it right or not?
Thank you in advance.

julien.decharentenay · February 8, 2011, 19:19

Are you talking about numerical diffusion/dispersion or the physical phenomena of diffusion/dispersion?

dut_thinker · February 8, 2011, 23:32

julien.decharentenay , thank you for your reply.
I am talking about phenomena of diffusion and dispersion?
By the way, can you also show me the difference of numerical diffusion and dispersion?

julien.decharentenay · February 9, 2011, 18:55

From the numerical point of view, numerical diffusion and dispersion reflect on the properties of the spatial discretisation employed:

- numerical diffusion indicates that the space discretisation operator will tend to smooth out sharp front/discontinuities, i.e. instead of having a sharp interface over 1 cell the space discretisation operator will spread it over a few cells;

- numerical dispersion refers to the properties of the space discretisation operator in not generating too high gradients, i.e. if you have a scalar between 0 and 1 with a sharp interface, the space discretisation operator will leads to value below 0 or exceeding 1.

From a practical point of view:
- an upwind discretisation scheme will have high numerical diffusion and low dispersion;
- a central or high order discretisation scheme (with no limiter) will have low numerical diffusion and high dispersion;
- a limited discretisation scheme tries to have the best of both world.

From a physical point of view, diffusion is the capacity of smoothing sharp interface: think you have two gas separated by a wall. Once you remove the wall, the two gas mixes by diffusion.

I am not too sure about dispersion.

Hope it helps. Julien

dut_thinker · February 9, 2011, 20:14

Julien de Charentenay
Thank you very much.
Your explanation really gave me a gret help.

hadian · February 10, 2011, 12:19

in mathematical view, numerical diffusion is created when the highest term in truncation error is a factor of even order difference ( $\frac{\partial ^2}{\partial x^2}$ و $\frac{\partial ^4}{\partial x^4}$ , ...) , however in numerical dispersion that is in odd order of difference ( $\frac{\partial ^3}{\partial x^3}$ و $\frac{\partial ^5}{\partial x^5}$ , ...).

dut_thinker · February 10, 2011, 20:11

Thanks for your reply, hadian.
Maybe it could be understood in this way. Just as you said, while the trunction error is even order difference, like diffusion equation it has diffusion characteristic; while the trunciton error is odd order difference, like advection equaiton, it will be propagated like wave.

Ahmed · February 10, 2011, 23:49

Quote:

Originally Posted by dut_thinker

According to my understanding, diffusion is a process caused by concentration difference. Dispersion is the combined effort of adveciton and diffusion.
Is it right or not?
Thank you in advance.

Your first post was not far from the physical meaning.
Diffusion is the transport of mass, energy, momentum as a result of the random Molecular movements, expressed in a mathematical language as the result of multiplying some constant by the first gradient of the quantity of interest, eg, heat conduction in solids results from multiplying the coefficient of thermal conductivity (k) by the first derivative of the driving force, temperature difference (dT/dx etc)
Dispersion is as you said, and the meaning of advection (convection) is clear
In numerical analysis, terms that look like their physical counter parts are called numerical ....................

dut_thinker · February 11, 2011, 00:09

Ahmed, thanks for your reply.
I totally agree with you about the defination of diffusion.
However, i don't think advection and convection are the same thing.
Because convectino equals to advection only while there is no diffusion.

MMGHEIJI · April 29, 2015, 10:22

Dear Hadian,
I guess you made a mistake in describing of EVEN-ODD ORDERS
Based on h.Jasak (1999), high resolution NVD differencing scheme for arbitrarily unstructured mesh paper, we would have a numerical diffusion if:
1.the highest of the truncation error includes ODD-ORDER spatial derivatives, the solution will be affected by a certain amount of numerical diffusion
2. If on the other hand, the leading truncation term include EVEN-ORDER spatial derivatives numerical dispersion occurs.

truffaldino · April 29, 2015, 12:57

Quote:

Originally Posted by dut_thinker

According to my understanding, diffusion is a process caused by concentration difference. Dispersion is the combined effort of adveciton and diffusion.
Is it right or not?
Thank you in advance.

And dispersion is a dependence of wave speed on their wavelength.

In the non-dispersive medium all the waves move with the same velocity.

Numerical diffusion is a particular case of that (dispersion happens due to presence of "higher derivatives").

February 7, 2011, 21:49	What is the difference between diffusion and dispersion?	#1
dut_thinker New Member Join Date: Feb 2010 Posts: 12 Rep Power: 16	According to my understanding, diffusion is a process caused by concentration difference. Dispersion is the combined effort of adveciton and diffusion. Is it right or not? Thank you in advance.

February 8, 2011, 19:19		#2
julien.decharentenay Senior Member Julien de Charentenay Join Date: Jun 2009 Location: Australia Posts: 231 Rep Power: 17	Are you talking about numerical diffusion/dispersion or the physical phenomena of diffusion/dispersion? __________________ --- Julien de Charentenay

February 9, 2011, 18:55		#4
julien.decharentenay Senior Member Julien de Charentenay Join Date: Jun 2009 Location: Australia Posts: 231 Rep Power: 17	From the numerical point of view, numerical diffusion and dispersion reflect on the properties of the spatial discretisation employed: - numerical diffusion indicates that the space discretisation operator will tend to smooth out sharp front/discontinuities, i.e. instead of having a sharp interface over 1 cell the space discretisation operator will spread it over a few cells; - numerical dispersion refers to the properties of the space discretisation operator in not generating too high gradients, i.e. if you have a scalar between 0 and 1 with a sharp interface, the space discretisation operator will leads to value below 0 or exceeding 1. From a practical point of view: - an upwind discretisation scheme will have high numerical diffusion and low dispersion; - a central or high order discretisation scheme (with no limiter) will have low numerical diffusion and high dispersion; - a limited discretisation scheme tries to have the best of both world. From a physical point of view, diffusion is the capacity of smoothing sharp interface: think you have two gas separated by a wall. Once you remove the wall, the two gas mixes by diffusion. I am not too sure about dispersion. Hope it helps. Julien rp.soni, subhankarnits and joshwilliams like this. __________________ --- Julien de Charentenay

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February 8, 2011, 23:32		#3
dut_thinker New Member Join Date: Feb 2010 Posts: 12 Rep Power: 16	julien.decharentenay , thank you for your reply. I am talking about phenomena of diffusion and dispersion? By the way, can you also show me the difference of numerical diffusion and dispersion?

February 9, 2011, 20:14		#5
dut_thinker New Member Join Date: Feb 2010 Posts: 12 Rep Power: 16	Julien de Charentenay Thank you very much. Your explanation really gave me a gret help.

February 10, 2011, 12:19		#6
hadian Member Mohammad Reza Hadian Join Date: Mar 2009 Location: Yazd, Iran Posts: 52 Rep Power: 17	in mathematical view, numerical diffusion is created when the highest term in truncation error is a factor of even order difference ( $\frac{\partial ^2}{\partial x^2}$ و $\frac{\partial ^4}{\partial x^4}$ , ...) , however in numerical dispersion that is in odd order of difference ( $\frac{\partial ^3}{\partial x^3}$ و $\frac{\partial ^5}{\partial x^5}$ , ...).

February 10, 2011, 20:11		#7
dut_thinker New Member Join Date: Feb 2010 Posts: 12 Rep Power: 16	Thanks for your reply, hadian. Maybe it could be understood in this way. Just as you said, while the trunction error is even order difference, like diffusion equation it has diffusion characteristic; while the trunciton error is odd order difference, like advection equaiton, it will be propagated like wave.

February 11, 2011, 00:09		#9
dut_thinker New Member Join Date: Feb 2010 Posts: 12 Rep Power: 16	Ahmed, thanks for your reply. I totally agree with you about the defination of diffusion. However, i don't think advection and convection are the same thing. Because convectino equals to advection only while there is no diffusion.

April 29, 2015, 10:22		#10
MMGHEIJI New Member Mohammadmahdi Ghiji Join Date: Jan 2010 Location: Australia Posts: 1 Rep Power: 0	Dear Hadian, I guess you made a mistake in describing of EVEN-ODD ORDERS Based on h.Jasak (1999), high resolution NVD differencing scheme for arbitrarily unstructured mesh paper, we would have a numerical diffusion if: 1.the highest of the truncation error includes ODD-ORDER spatial derivatives, the solution will be affected by a certain amount of numerical diffusion 2. If on the other hand, the leading truncation term include EVEN-ORDER spatial derivatives numerical dispersion occurs.