We have the following partial differential equation
$$u_t =H(u)_{xx},~~~ 0\le x<1$$
with an initial condition $u(x,0) = f(x)$ and periodic boundary condition. Here $0 \le H'(u) \le d$. Consider the following one-step, three-point scheme on a uniform mesh $x_j = j\Delta x$ with spatial mesh size $\Delta x$:
$$u_j^{n+1} =u_j^n +aH(u_{j-1}^n)+bH(u_j^n)+cH(u_{j+1}^n),$$
where $a, b, c$ are constants which may depend on the mesh ratio $\mu = \Delta t/\Delta x^2$, $\Delta t$ is the time step, and $u_j^n$ approximates the exact solution at $u(x_j, t^n)$ with $t^n = n\Delta t$.
Find the CFL number $\mu_0$ such that the above scheme, with the constants determined that the above scheme is second order accurate, is stable under the time step restriction $\mu \le\mu_0$. Please specify which norm you are using for stability, and prove this stability result.
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Basic calculation shows that $a=\mu,b=-2\mu,c=\mu$. Since here we have an another function $H$ with $H'$ bounded (I think Von-Neumann analysis no longer works here), I think here we should choose $L_\infty$ norm instead of $L_2$ norm. However, I don't know how to use the condition $0 \le H'(u) \le d$. We see $u_j^{n+1}-u_j^n=\frac{\Delta t}{2}H(u_j^n)xx+O(\Delta t\Delta h^2$), but the expansion of $H(u_j^n)xx$ consists many terms and I don't know how to deal with it. Maybe this way of approaching is wrong. Is there any possible approaching method/solutions to this problem? Thanks!
Best Answer
You get a linear system $$\vec u^{n+1}=\vec u^n+Δt·A·\vec{H}(\vec u^n)$$ which is the explicit Euler method for $$\dot {\vec u}=A·\vec H(\vec u).$$ A Lipschitz constant for the right side based on max or $\ell^1$ vector norms is $$L=(|a|+|b|+|c|)\frac{d}{Δt}.$$ Stability for the Euler method requires $$ LΔt\le 2 $$ with, as you computed, $LΔt=4μd$. Thus one should have $μ\le \frac{1}{2d}$.