Developing a finite difference scheme for non-linear conservation law

Question

When considering equations of the form $$u_t + f(u)_x = 0, $$ there are many finite difference schemes that will converge to smooth solutions, such as the one based on Lax-Friedrichs, $$\frac{u_{j}^{n+1} - \frac{1}{2}(u_{j+1}^n + u_{j-1}^n)}{k} + \frac{f(u_{j+1}^n)-f(u_{j-1}^n)}{2h} = 0.$$ This scheme in particular is consistent, TV-stable, and the flux function $$F(v,w) = \frac{\lambda}{2}(w-v) + \frac{1}{2}(f(v)+f(w)) $$ is Lipschitz, so the numerical solution will converge to the weak solution as $h, k \to 0$. However, I believe schemes like these may converge to weak solutions that violate the entropy condition, or are only valid for a finite time $T$. Does there exist a generalization to these methods that converge for all $t > 0$? I would imagine we could consider schemes involving an artificial viscosity term $\epsilon u_{xx}$, since we would expect smooth solutions for $t >0$, and take the limit as $\epsilon \to 0$ to get our desired approximation. Any help would be greatly appreciated.

Answer 1

Under the Courant-Friedrichs-Lewy condition, the Lax-Friedrichs method converges to the vanishing viscosity solution by construction. It is one of the most robust methods, and will provide entropy solutions.

Even if that isn't a proof of the result you are looking for, let's look for the scheme's modified equation (see e.g. literature references by R.J. LeVeque). Using Taylor series, one shows that for smooth solutions \begin{aligned} \frac{u_j^{n+1} - \frac12( u_{j+1}^n + u_{j-1}^n )}{k} &= (u_t)_j^n + \frac{k}{2}(u_{tt})_j^n - \frac{h^2}{2k} (u_{xx})_j^n + O(h^2, k^2) \\ \frac{f(u_{j+1}^{n}) - f(u_{j-1}^n)}{2h} &= [f(u)_x]_j^n + O(h^2) . \end{aligned} Up to higher-order terms (h.o.), we therefore have $u_t + f(u)_x = \frac{h}{2\lambda} (u_{xx} - \lambda^2 u_{tt})$ with $\lambda = k/h$. Then, the PDE produces \begin{aligned} u_{tx} &= -f(u)_{xx} = -f'(u) u_{xx} \\ u_{tt} &= -f(u)_{xt} = -f'(u)u_{xt}, \end{aligned} which leads to the scheme's modified equation $$ u_t + f(u)_x = \frac{h}{2\lambda}\left(1 - \lambda^2 f'(u)^2\right) u_{xx} + \text{h.o.} $$ Note the vanishing diffusion term in the r.h.s.