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August 29, 2011, 20:35 
Jacobian of 1D Euler Equation of Gas Dynamics

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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! 

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September 1, 2011, 23:33 

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Bidyut
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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

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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 

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irfan khan
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Quote:
It is simply rate of change of "numerical flux" with respect to "quasi" primitive variable rho, pho*u and e... 

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