Euler Equations (fluid Dynamics) - Non-conservation Form With Flux Jacobians

Non-conservation Form With Flux Jacobians

Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. From the original equations as given above in vector and conservation form, the equations are written in a non-conservation form as:

$\frac{\partial \bold m}{\partial t} + \bold A_x \frac{\partial \bold m}{\partial x} + \bold A_y \frac{\partial \bold m}{\partial y} + \bold A_z \frac{\partial \bold m}{\partial z} = {\bold 0}.$

where A_x, A_y and A_z are called the flux Jacobians, which are matrices equal to:

$\bold A_x=\frac{\partial \bold f_x(\bold s)}{\partial \bold s}, \qquad \bold A_y=\frac{\partial \bold f_y(\bold s)}{\partial \bold s} \qquad \text{and} \qquad \bold A_z=\frac{\partial \bold f_z(\bold s)}{\partial \bold s}.$

Here, the flux Jacobians A_x, A_y and A_z are still functions of the state vector m, so this form of the Euler equations is nonlinear, just like the original equations. This non-conservation form is equivalent to the original Euler equations in conservation form, at least in regions where the state vector m varies smoothly.

Read more about this topic: Euler Equations (fluid Dynamics)

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