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I've been discretizing PDEs and formulating $Ax=b$ systems, and yet I don't really know what the $A$ and $b$ are in words.

I occasionally call the $A$ matrix the "Jacobian matrix," but for linear PDEs, "Jacobian" doesn't make sense, so I call it the "differential operator matrix." Is there a standard name for this $A$ matrix?

For the $b$ vector, I call it the forcing vector, but I don't know if that is correct either.

Are there standard names for both of these variables after applying a discretization to a PDE (whether linear or nonlinear)?

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$A$ is a discretized version of your differential operator + enforced boundary conditions. The names for $A$ can vary depending on the way the PDE is being discretized. For example, in FEM, it will be stiffness matrix. For integral equation methods (technically not a PDE) applied to Maxwell equations, such a matrix is usually called impedance matrix.

However, I assume that you are using finite differences to discretized PDE. I don't know a special name, but I've seen in multiple papers such notions as finite difference matrix, discretized operator, etc.

For the RHS, forcing vector is something that I've seen a lot around; though, usually in the meaning of the forcing function that produces the RHS in a discretized form. Another way to name $b$ is excitation, which is more intuitive, as it describes the way the problem is setup and "excited" (sorry for the tautology).

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  • $\begingroup$ Ah I have never heard of $b$ referred to as "excitation." I am generally using FVM, but I assume the discretized operator terminology still applies to FVM? In retrospect, it seems most people verbally refer to the $A$ matrix as the $A$ matrix and don't really give it a name. $\endgroup$ – user27504 May 11 '18 at 2:51
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    $\begingroup$ @user27504 yep, I would say that for this terminology FVM should be similar to FD. $\endgroup$ – Anton Menshov May 11 '18 at 3:02
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Here is some generic (application-independent) terminology I've seen in papers:

  • $A$: The ``coefficient matrix''

  • $b$: The ``right hand side''

  • $x$: The ``unknown''

Also, sometimes $A$ may be called the ``coefficient operator'' if it is considered as a linear operator rather than a matrix.

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