# Von Neumann stability analysis in 3d

I need to get a stability criterion for the numerical scheme for equation

$$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 u}{\partial z^2} -\alpha\frac{\partial u}{\partial x}-\beta\frac{\partial u}{\partial y}-\gamma\frac{\partial u}{\partial z}-f=0$$

The scheme is explicit: $$\frac{u^{n+1}_{i,j,k}-u^{n}_{i,j,k}}{\Delta t}=\frac{u^{n}_{i+1,j,k}-2u^{n}_{i,j,k}+u^{n}_{i-1,j,k}}{\Delta x^2}+\frac{u^{n}_{i,j+1,k}-2u^{n}_{i,j,k}+u^{n}_{i,j-1,k}}{\Delta y^2}+$$ $$+\frac{u^{n}_{i,j,k+1}-2u^{n}_{i,j,k}+u^{n}_{i,j,k-1}}{\Delta z^2}+\alpha\frac{u^{n}_{i+1,j,k}-u^{n}_{i-1,j,k}}{2\Delta x }+$$ $$+\beta\frac{u^{n}_{i,j+1,k}-u^{n}_{i,j-1,k}}{2\Delta y }+\gamma\frac{u^{n}_{i,j,k+1}-u^{n}_{i,j,k-1}}{2\Delta z }+f_{i,j,k}.$$ After that I found an expression with a help of von Neumann analysis for amplification coefficient:

$$\rho=1-4\Delta t\left(\frac{\sin^2{\theta /2}}{\Delta x^2}+\frac{\sin^2{\eta /2}}{\Delta y^2}+\frac{\sin^2{\zeta /2}}{\Delta z^2}\right)+i\Delta t\left(\alpha\frac{\sin{\theta}}{\Delta x}+\beta\frac{\sin{\eta}}{\Delta y}+\gamma\frac{\sin{\zeta }}{\Delta z}\right)$$

To find a condition for stability criterion I must find a condition for $\Delta t,\Delta x,\Delta y,\Delta z$ so that for all $\theta,\eta,\zeta$ an inequality $|\rho|<1$ is true. Here I stuck.Could you help me with this stuff?

• What expression do you obtain when you substitute an arbitrary fourier mode? – Paul Mar 10 '14 at 17:21
• @Paul (who asked the question) you should put your comment into the body of the question. Then look for conditions under which $|\rho|\le 1$. – David Ketcheson Mar 11 '14 at 4:18

## 1 Answer

Though your problem is in 3 dimensions, you can gain a lot of insight by considering the 1D case. Let's omit the y and z terms and concentrate on x and t. Then, your problem would be equivalent to the classic advection-diffusion equation with an explicit Forward-in-Time, Centered-in-Space (FTCS) finite difference method.

Knowing the name of the problem and the method, it's fairly easy to find derivations online. I personally found this stability analysis for the advection diffusion equation particularly insightful. Following this approach, you should be able to extrapolate a method for the 3D case.