Jacobian contributions in linearized residual equation

uni · November 16, 2018, 15:12

Hi, I've been trying to figure out how equation (45) was derived in the AIAA 2013-0287 paper by Palacios et al. https://www.google.com/url?sa=t&rct=...416y4HWs2meEMz

It follows equation (44) where the linear residual system is solved to find $\Delta U_i^n$. The part of the paper I'm stuck at is shown in the figure attached.

Can someone please tell me how to derive this?

hlk · November 16, 2018, 21:29

I think that rather than being a derivation step, this is a statement of a definition, that the contribution to the the jacobian of a flux vector that is dependent on two points (i,j), only adds terms at the locations as indicated.

In other words, (45) is included as additional detail to help understand the general nature of the jacobian, not as a derivation step that follows (44).

You may already know this, but just in case, the ":=" symbol is used to specify a definition rather than an equation.

uni · November 17, 2018, 10:57

Okay then, It makes sense. The contributions are the same but opposite signs at nodes i and j. This is also what is done in all the ::ComputeResidual* functions in the code.

Thanks

November 16, 2018, 21:29		#2
hlk Senior Member Heather Kline Join Date: Jun 2013 Posts: 309 Rep Power: 13	I think that rather than being a derivation step, this is a statement of a definition, that the contribution to the the jacobian of a flux vector that is dependent on two points (i,j), only adds terms at the locations as indicated. In other words, (45) is included as additional detail to help understand the general nature of the jacobian, not as a derivation step that follows (44). You may already know this, but just in case, the ":=" symbol is used to specify a definition rather than an equation. aero_amit and saladbowl like this.

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November 17, 2018, 10:57		#3
uni New Member Uni Join Date: Mar 2018 Location: Panama Posts: 4 Rep Power: 8	Okay then, It makes sense. The contributions are the same but opposite signs at nodes i and j. This is also what is done in all the ::ComputeResidual* functions in the code. Thanks