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Simplicity. August 29, 2011 20:35

Jacobian of 1D Euler Equation of Gas Dynamics
 
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_j-1) + 0.5*(dt/dx)*a*(p_j+1 -2p_j + p_j-1).

Any suggestion/idea is welcome and appreciated!

bidyut September 1, 2011 23:33

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 Quasi-linear 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 time-integration 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.

Simplicity. September 2, 2011 04:09

Jacobian of 1D Euler Equation of Gas Dynamics
 
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.

imbaasat September 6, 2011 14:12

Quote:

Originally Posted by Simplicity. (Post 322614)
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.

Here Jacobian is "not" transformation from one to other coordinate system.
It is simply rate of change of "numerical flux" with respect to "quasi" primitive variable rho, pho*u and e...


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