- #1
liquidFuzz
- 97
- 3
I'm trying to refresh some numerical science stuff. Von Neumann analysis, if I take a slimmed down equation, convection. [tex] \frac{∂u}{∂t}+a ∇ u =0[/tex] If I'm using Euler forward, [tex] \frac{u^{n+1}-u^n}{\Delta t}+\frac{a}{2h} \left( u_{j+1}^n -u_{j-1}^n \right) =0[/tex] For [itex]\hat{u}^n = G^n\hat{u}^0[/itex] a growth factor [itex]|G|\leq1[/itex] is sufficient. This gives, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \beta h \left( \frac{u_{j+1}^n-2u_j^n + u_{j-1}^n}{h^2} \right)[/tex] With the smallest possible diffusion but still stable, [itex]\beta \geq \frac{a}{2}[/itex] becomes [itex]\beta = \frac{a}{2}[/itex]. If this is plugged into the equation, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \frac{a}{2h} \left( {u_{j+1}^n-2u_j^n + u_{j-1}^n} \right)[/tex] Now, according to some old notes I have the result is, [tex] \frac{u^{n+1}-u^n}{\Delta t} + \frac{1}{h} \left( {au_j^n -a u_{j-1}^n} \right)[/tex] Well this is somewhat vague for me. Specially the last step where the defraction term is finalised.
Any pointers of help would be appreciated, specially that last step.
Any pointers of help would be appreciated, specially that last step.