# Adding a diffusion term to the MUSCL - Kurganov and Tadmor central scheme

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:

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

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

• Note that your material parameter is missing in your discretization. Nov 19, 2023 at 12:39