
[Sponsors] 
Jacobian of 1D Euler Equation of Gas Dynamics 

LinkBack  Thread Tools  Search this Thread  Display Modes 
August 29, 2011, 20:35 
Jacobian of 1D Euler Equation of Gas Dynamics

#1 
New Member
Join Date: Aug 2011
Posts: 4
Rep Power: 14 
I'm new in the field of CFD and now writing an optimization code that incorporates 1D system of Euler equations of Gas Dynamics.
Some lines in my code need Jacobian of the Euler equation. I wonder how to incorporate Jacobian, because to my knowledge, for 1D Euler equation, Jacobian is a 3X3 matrix while my code uses one dimensional vectors/arrays? See for example discrete equation below where f'(u_j) is the Jacobian. p_i = p_j + (dt/dx)*q_j*f'(u_j) + 0.5*(dt/dx)*(q_j+1  q_j1) + 0.5*(dt/dx)*a*(p_j+1 2p_j + p_j1). Any suggestion/idea is welcome and appreciated! 

September 1, 2011, 23:33 

#2 
New Member
Bidyut
Join Date: Sep 2011
Posts: 1
Rep Power: 0 
1 D Euler equations are of the form U_t + (F(U))_x=0, where A=dF/dU is the Jacobian, Now if you write this equation in Quasilinear form, then it becomes U_t + (A) U_x =0. U is a column vector of 3 elements namely rho, rho*u and e, so solve it using any scheme, U is 3X1 and A is 3x3 matrix. So ultimately there will be three coupled equations which you need to solve simultaneously by continuous updation of rho, rho*u and e, and also dt, using any timeintegration method.
For a good reference, look into the paper : Sod, G. A. (1978). "A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws". J. Comput. Phys. 27: 1–31. 

September 2, 2011, 04:09 
Jacobian of 1D Euler Equation of Gas Dynamics

#3 
New Member
Join Date: Aug 2011
Posts: 4
Rep Power: 14 
Thank you bidyut, that is helpful enough. But I wonder if it is Jacobian that is used in the scheme or it just stands for some physical meaning.
Last edited by Simplicity.; September 2, 2011 at 04:32. 

September 6, 2011, 14:12 

#4  
New Member
irfan khan
Join Date: Aug 2011
Posts: 11
Rep Power: 14 
Quote:
It is simply rate of change of "numerical flux" with respect to "quasi" primitive variable rho, pho*u and e... 

Thread Tools  Search this Thread 
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Calculation of the Governing Equations  Mihail  CFX  7  September 7, 2014 06:27 
Open boundary condition for 1d euler equation?  Ashley  Main CFD Forum  3  August 3, 2009 04:05 
Do U have this book 'Gas dynamics by Rathakrishnan  Brian  Main CFD Forum  1  November 9, 2003 23:30 
Rarefied Gas Dynamics  Andrea Panizza  Main CFD Forum  3  July 15, 2003 07:45 
Gas pressure question  Dan Moskal  Main CFD Forum  0  October 24, 2002 22:02 