Method of characterestics (MOC)

Farouk · July 4, 2013, 11:56

Hi, I am currently trying to understand the position and state diagrams used in the Method Of Characteristics (used to solve PDE for CFD)...I can't find much useful literature about the topic..so if any one has suggestions, that would be of great help.
Thanks in advance.

FMDenaro · July 4, 2013, 12:34

Quote:

Originally Posted by Farouk

Hi, I am curently trying to understand the position and state diagrams used in the Method Of Characterestics (used to solve PDE for CFD)...I can't find much usefull literature about the topic..so if any one has suggestions, that would be of great help.
Thanks in advance.

see the classical book of Zucrow

Farouk · July 7, 2013, 12:25

Quote:

Originally Posted by FMDenaro

see the classical book of Zucrow

Hi FMDenaro and thanks for the reply.
The text you suggested was very helpful, but still, it didn't answer all my questions, mainly, the use of left going $\lambda$ and right going $\beta$ characteristics.
For those who are interested, you can see below the two diagrams for a simple wave.
Does anyone happen to have a practical explanation?
Thanks

FMDenaro · July 7, 2013, 12:38

have a look to the simple wave analysis, homoentropic flows have two Riemann invariants, one along each characteristic curve that allow you to get an exact solution. Generally, the application is reported as a piston moving in a tube without dissipation. The book of Zucrow illustrated such case. Furthermore, the more modern book of LeVeque can be also useful

Jonny6001 · July 8, 2013, 16:57

Hello.

Generally speaking the method of characteristics is used to change a PDE to an ODE. It does this by making one of the variables a function of the other variable.
I like to think of the method by considering a second coordinate system that moves along the second variable. It's position along the second variable is a function of the first variable. Then say that the derivative of the function with respect to the new coordinate system is 0. Therefore the characteristic must stay constant along the curves.

This is a very simplified analogy but here goes!

On the 'x-t' diagram with t=0 each characteristic curve has a value. The gradient of the curves emerging from the t=0 represents the speed of information propagation. Think of it as several cars starting from different positions at the same time. If all cars move parallel to each other (have the same velocity) they will never hit each other. However if the car behind moves quicker (has a different gradient to the characteristic curve) it will eventually hit the car in front. This is the point of a shockwave.

The method can be used for the exact solution of equations such as linear advection. A non-linear equation (Burgers) results in curved characteristic curves.

Hope that's not confused you further.

Farouk · July 9, 2013, 03:16

Quote:

Originally Posted by Jonny6001

Hello.

Generally speaking the method of characteristics is used to change a PDE to an ODE. It does this by making one of the variables a function of the other variable.
I like to think of the method by considering a second coordinate system that moves along the second variable. It's position along the second variable is a function of the first variable. Then say that the derivative of the function with respect to the new coordinate system is 0. Therefore the characteristic must stay constant along the curves.

This is a very simplified analogy but here goes!

On the 'x-t' diagram with t=0 each characteristic curve has a value. The gradient of the curves emerging from the t=0 represents the speed of information propagation. Think of it as several cars starting from different positions at the same time. If all cars move parallel to each other (have the same velocity) they will never hit each other. However if the car behind moves quicker (has a different gradient to the characteristic curve) it will eventually hit the car in front. This is the point of a shockwave.

The method can be used for the exact solution of equations such as linear advection. A non-linear equation (Burgers) results in curved characteristic curves.

Hope that's not confused you further.

Hi Jonny6001 and thanks for the answer. That's exactly the type of answers I was looking for (no mathematics, but logic)...I will try to work some theory and examples, hopefully, I will manage to implement something simple, and i will post back to this thread.

Cheers.

Farouk · July 9, 2013, 09:16

Hi there, does any one happen to have an implemented MOC code? possibly in Matlab? That would be a good start for me.

Regards

Yashao · January 15, 2015, 16:23

I'm studying the 3d MOC in reacting flow recently, I read several papers and books and now I understand the characteristic surfaces, and I can get the the compatibility equation, but the following numerical part really makes me headache. How to solve the compatibility equation in finite difference is very hard for me.
I'd appreciate it if anyone can give me some suggestion!

July 4, 2013, 11:56	Method of characterestics (MOC)	#1
Farouk New Member CFDLearner Join Date: Jul 2013 Posts: 16 Rep Power: 12	Hi, I am currently trying to understand the position and state diagrams used in the Method Of Characteristics (used to solve PDE for CFD)...I can't find much useful literature about the topic..so if any one has suggestions, that would be of great help. Thanks in advance. Last edited by Farouk; July 7, 2013 at 12:27.

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July 7, 2013, 12:38		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71	have a look to the simple wave analysis, homoentropic flows have two Riemann invariants, one along each characteristic curve that allow you to get an exact solution. Generally, the application is reported as a piston moving in a tube without dissipation. The book of Zucrow illustrated such case. Furthermore, the more modern book of LeVeque can be also useful

July 8, 2013, 16:57		#5
Jonny6001 Member Jonny Join Date: Aug 2009 Posts: 72 Rep Power: 16	Hello. Generally speaking the method of characteristics is used to change a PDE to an ODE. It does this by making one of the variables a function of the other variable. I like to think of the method by considering a second coordinate system that moves along the second variable. It's position along the second variable is a function of the first variable. Then say that the derivative of the function with respect to the new coordinate system is 0. Therefore the characteristic must stay constant along the curves. This is a very simplified analogy but here goes! On the 'x-t' diagram with t=0 each characteristic curve has a value. The gradient of the curves emerging from the t=0 represents the speed of information propagation. Think of it as several cars starting from different positions at the same time. If all cars move parallel to each other (have the same velocity) they will never hit each other. However if the car behind moves quicker (has a different gradient to the characteristic curve) it will eventually hit the car in front. This is the point of a shockwave. The method can be used for the exact solution of equations such as linear advection. A non-linear equation (Burgers) results in curved characteristic curves. Hope that's not confused you further.

July 9, 2013, 09:16		#7
Farouk New Member CFDLearner Join Date: Jul 2013 Posts: 16 Rep Power: 12	Hi there, does any one happen to have an implemented MOC code? possibly in Matlab? That would be a good start for me. Regards

January 15, 2015, 16:23		#8
Yashao New Member yang Shao Join Date: Jan 2015 Posts: 3 Rep Power: 11	I'm studying the 3d MOC in reacting flow recently, I read several papers and books and now I understand the characteristic surfaces, and I can get the the compatibility equation, but the following numerical part really makes me headache. How to solve the compatibility equation in finite difference is very hard for me. I'd appreciate it if anyone can give me some suggestion!