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Im currently using a MUSCL scheme with a rusanov flux and Van Leer limiter to simulate the 2d euler equations:

$$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} + \frac{\partial \rho v_y}{\partial y} = 0 $$

$$ \frac{\partial \rho v_x}{\partial t} + \frac{\partial (\rho v_x^2 + P)}{\partial x} + \frac{\partial (\rho v_y v_x)}{\partial y} = 0 $$

$$ \frac{\partial \rho v_x}{\partial t} + \frac{\partial (\rho v_x v_y)}{\partial x} + \frac{\partial (\rho v_y^2 + P)}{\partial y} = 0 $$

$$ \frac{\partial E}{\partial t} + \frac{\partial ((E +P)v_x)}{\partial x} + \frac{\partial ((E +P)v_y)}{\partial y} = 0 $$

Where $\rho$ is mass density, $v_{x,y}$ is the x and y fluid velocity, $E$ is total internal energy density, and $P$ is thermal pressure (pressure is closed via: $P = (\gamma -1) (E - \frac{\rho v_x^2}{2} - \frac{\rho v_y^2}{2})$). Specifically, I am using the Kurganov and Tadmor central scheme (KT Scheme) (https://en.wikipedia.org/wiki/MUSCL_scheme) to calculate the fluxes at the faces.

I would like to add $\eta \nabla^2v_x$ and $ \eta \nabla^2 v_y$ terms to the momentum equations such that the momentum density equations are now:

$$ \frac{\partial \rho v_x}{\partial t} + \frac{\partial (\rho v_x^2 + P)}{\partial x} + \frac{\partial (\rho v_y v_x)}{\partial y} = \eta (\frac{\partial^2 v_x}{\partial x^2} + \frac{\partial^2 v_x}{\partial y^2}) $$

$$ \frac{\partial \rho v_x}{\partial t} + \frac{\partial (\rho v_x v_y)}{\partial x} + \frac{\partial (\rho v_y^2 + P)}{\partial y} = \eta (\frac{\partial^2 v_y}{\partial x^2} + \frac{\partial^2 v_y}{\partial y^2}) $$

If I understand the KT scheme correctly, I can add this diffusion terms and keep the scheme explicit. Following the formula given on wikipedia, would the discretization of the diffusion terms be:

$$ \eta \nabla^2v_x = \frac{v_x^{i+1,j} - 2v_x^{i,j} + v_x^{i-1,j}}{(\Delta x_i)^2} + \frac{v_x^{i,j+1} - 2v_x^{i,j} + v_x^{i,j-1}}{(\Delta y_i)^2}$$

$$ \eta \nabla^2v_y = \frac{v_y^{i+1,j} - 2v_y^{i,j} + v_y^{i-1,j}}{(\Delta x_i)^2} + \frac{v_y^{i,j+1} - 2v_y^{i,j} + v_y^{i,j-1}}{(\Delta y_i)^2}$$?

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  • $\begingroup$ Note that your material parameter is missing in your discretization. $\endgroup$
    – ConvexHull
    Nov 19, 2023 at 12:39

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From a finite volume point of view, fluxes should be calculated at the cell faces and added up for each cell.

However, since you are using Cartesian meshes and your material properties are constant, you can choose a straightforward central (finite difference) approximation, as given in your example.

Note that your time step will be more restrictive including parabolic effects.

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