Question about characteristics and classification of second-order PDEs

Time4Tea · November 28, 2018, 15:03

Hi, I am currently reading the text book 'Computational Techniques for Fluid Dynamics', by C.A.J. Fletcher; however, I am confused about a section in Chapter 2 relating to classification and solving of second-order PDEs using the method of characteristics.

I have posted the details of the question here on the site Mathematics Stack Exchange, so I will not repeat the mathematical details in this post.

I posted the question almost two weeks ago and no-one there has yet provided an answer. I am just wondering if anyone on this forum can help shed some light on this, either by posting something here or directly on that site?

Thanks in advance!

FMDenaro · November 28, 2018, 15:17

Quote:

Originally Posted by Time4Tea

Hi, I am currently reading the text book 'Computational Techniques for Fluid Dynamics', by C.A.J. Fletcher; however, I am confused about a section in Chapter 2 relating to classification and solving of second-order PDEs using the method of characteristics.

I have posted the details of the question here on the site Mathematics Stack Exchange, so I will not repeat the mathematical details in this post.

I posted the question almost two weeks ago and no-one there has yet provided an answer. I am just wondering if anyone on this forum can help shed some light on this, either by posting something here or directly on that site?

Thanks in advance!

I never followed Fletcher in the classification, I largely prefer the wave solution technique described in the book of Hirsch.
However, I think that Eq.(4) is assumed to be satisfied if both terms in the LHS of Eq.(4) are zero

Time4Tea · November 28, 2018, 18:30

@FMDenaro: Thanks, I will try to get hold of a copy of that textbook, to see if I will understand the explanation there better.

I guess I can also just skip past that chapter in Fletcher and go on to the sections that deal more with the numerical methods. However, in general, I don't like skipping past sections of text books without understanding them. It seems to me that this chapter of the Fletcher book is a bit poorly-written, if it doesn't explain the material well enough to allow someone to answer the problems at the end. It's a bit of a pet peeve of mine, oh well ...

FMDenaro · November 28, 2018, 18:45

This topic is just of mathematical character, not necessarily linked to CFD so any good textbook on PDE can be used, too.

However, among many CFD textbooks you could find useful a reading to Chap.3 of Anderson

agd · November 29, 2018, 11:59

There is a better discussion in the book "Computational Fluid Mechanics and Heat Transfer" by Anderson, Tannehill, and Pletcher that follows a similar development but is more clearly explained. The equation you note in the StackExchange post arises as the determinant of the matrix of coefficients that comes from the 3x3 system developed from the PDE and the two equations for the derivatives of p and q in terms of u, v, w, and the characteristic curve arc parameter tau. No mention is made of splitting the original equation.

praveen · November 30, 2018, 08:02

I like Whitham's definition of pde types. I dont have the book with me but I highly recommend looking it up. Hyperbolic equations have wave-like solutions. Parabolic equations have damped wave-like solutions.

We better understand canonical equations like

$u_t = u_{xx} \qquad (parabolic)$

$u_{tt} = u_{xx} \qquad (hyperbolic)$

$u_{xx} + u_{yy} = 0 \qquad (elliptic)$

If given a general second order PDE, then the different conditions on the coefficients in the PDE
$B^2 - 4AC > 0, \quad \textrm{or} \quad = 0 \quad \textrm{or} \quad < 0$
allow you to find a change of independent variables that tranforms the PDE into one of the above three canonical PDE. I find this view more useful for classification of pde.

agd · November 30, 2018, 13:56

I would recommend Whitham's entire book "Linear and Non-linear Waves".

November 28, 2018, 15:03	Question about characteristics and classification of second-order PDEs	#1
Time4Tea Senior Member Lee Strobel Join Date: Jun 2016 Posts: 133 Rep Power: 10	Hi, I am currently reading the text book 'Computational Techniques for Fluid Dynamics', by C.A.J. Fletcher; however, I am confused about a section in Chapter 2 relating to classification and solving of second-order PDEs using the method of characteristics. I have posted the details of the question here on the site Mathematics Stack Exchange, so I will not repeat the mathematical details in this post. I posted the question almost two weeks ago and no-one there has yet provided an answer. I am just wondering if anyone on this forum can help shed some light on this, either by posting something here or directly on that site? Thanks in advance!

November 30, 2018, 08:02		#6
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 343 Blog Entries: 6 Rep Power: 18	I like Whitham's definition of pde types. I dont have the book with me but I highly recommend looking it up. Hyperbolic equations have wave-like solutions. Parabolic equations have damped wave-like solutions. We better understand canonical equations like $u_t = u_{xx} \qquad (parabolic)$ $u_{tt} = u_{xx} \qquad (hyperbolic)$ $u_{xx} + u_{yy} = 0 \qquad (elliptic)$ If given a general second order PDE, then the different conditions on the coefficients in the PDE $B^2 - 4AC > 0, \quad \textrm{or} \quad = 0 \quad \textrm{or} \quad < 0$ allow you to find a change of independent variables that tranforms the PDE into one of the above three canonical PDE. I find this view more useful for classification of pde. __________________ http://cpraveen.github.io http://twitter.com/cfdlab http://github.com/cpraveen

November 28, 2018, 18:30		#3
Time4Tea Senior Member Lee Strobel Join Date: Jun 2016 Posts: 133 Rep Power: 10	@FMDenaro: Thanks, I will try to get hold of a copy of that textbook, to see if I will understand the explanation there better. I guess I can also just skip past that chapter in Fletcher and go on to the sections that deal more with the numerical methods. However, in general, I don't like skipping past sections of text books without understanding them. It seems to me that this chapter of the Fletcher book is a bit poorly-written, if it doesn't explain the material well enough to allow someone to answer the problems at the end. It's a bit of a pet peeve of mine, oh well ...

November 28, 2018, 18:45		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	This topic is just of mathematical character, not necessarily linked to CFD so any good textbook on PDE can be used, too. However, among many CFD textbooks you could find useful a reading to Chap.3 of Anderson

November 29, 2018, 11:59		#5
agd Senior Member Join Date: Jul 2009 Posts: 358 Rep Power: 19	There is a better discussion in the book "Computational Fluid Mechanics and Heat Transfer" by Anderson, Tannehill, and Pletcher that follows a similar development but is more clearly explained. The equation you note in the StackExchange post arises as the determinant of the matrix of coefficients that comes from the 3x3 system developed from the PDE and the two equations for the derivatives of p and q in terms of u, v, w, and the characteristic curve arc parameter tau. No mention is made of splitting the original equation.

November 30, 2018, 13:56		#7
agd Senior Member Join Date: Jul 2009 Posts: 358 Rep Power: 19	I would recommend Whitham's entire book "Linear and Non-linear Waves".