For linear problems we can deduce convergence of numerical schemes by checking consistency and stability because of the Lax Equivalence Theorem.

For conservation laws, we know that conservative, consistent and monotone schemes will converge to the correct entropy solution (I know there are better schemes).

But do we have any 'simple' results for equations like the viscous Burgers' equation: $$ u_t+\left(\frac{u^2}{2}\right)_x=c u_{xx} $$ that yield some form of convergence? By 'simple' I just mean relatively easy to check, like the first two scenarios above.

Or failing that, do we just do 'heuristic' stability checks similar to using von Neumann analysis on frozen coefficient problems?