Stability analysis simplification for PDE

Question

I have the nonlinear PDE $$\frac{\partial U(z,t)}{\partial t} + A(U)\frac{\partial U(z,t)}{\partial z} + B(U)U(z,t) + C(z,t) = 0,$$ where $A(U)$ and $B(U)$ are guaranteed to be real and positive.

I want to use the MacCormack finite difference method to numerically solve it. For (Von Neumann) stability analysis, I use a linearized version of the PDE without $C(z,t)$, as that term will not contribute to instability in the error.

Without $B$, I retrieve the expected C.F.L. condition. That case is also a typical example found in texts on the MacCormack method and its stability, but I have not seen any examples where $B \neq 0$. If I don't neglect $B$, the stability analysis is still doable, just more messy than without it and the resulting condition on $\Delta t$ (and $\Delta z$) is less obvious.

So, is it safe to neglect $B$ for simplicity in the stability analysis?

Answer 1

Consider the case where $A=C=0$ and, as you stated, $B>0$. Then you simply have an ODE at every point $z$ with a solution that decays to zero (at least if $B$ is bounded away from zero). You can then apply the usual step length criteria developed for ODEs. In other words, you will need $$ \Delta t \le \frac{c}{B(z,t)} $$ if you were to use an explicit method (e.g., the forward Euler method, for which $c=2$; other methods have other stability constants) for the ODE at $z$. Since you want to use the same time step everywhere, you need $$ \Delta t \le c \min_{z\in\Omega} \frac{1}{B(z,t)}. $$