regularity of a solution and its affect on the global error

Question

I am solving the following equation $u_t=x^2u_{xx}+\frac{x-y}{T-t}u_y$ with an initial data $u_0=max(y-C,0)$ for some $C$ in the domain of the numerical method. In time I am solving that on $[0,T]$. Before I implement a numerical method I have to make sure the solution exist and unique. Assume it is done and the next question is the regularity since the error analysis is based on the Taylor expansion and to do so I need enough regularity in the solution. So, I would like to know what I can have in this case?

In $y$ dimension this equation is just transport and thus this non differentiability might propagate, but does the diffusion in the first variable help to smooth it at all? If not, then I don't even have $u_y$ in classical sense and only in weak, so why any numerical method would ever converge because if the local truncation error includes $u_{yy}$ or any higher order terms I can never have boundedness of it and at the point of the kink, and thus no mesh refinement can achieve decreasing error at that bad point.

Thanks!

Answer 1

Your diffusion only acts in $x$ direction and so, consequently, any kinks in the solution will only be smoothed away if the are along lines in $x$ direction. But, in your case, the discontinuity is only in $y$ direction, and consequently, the kink will never disappear -- it will only be transported. The easiest way to see this is to think of the solution as the limit $\Delta t \rightarrow 0$ of an operator splitting approach in which you first apply the diffusion step in $x$ direction and then the transport step in $y$ direction.

Based on this hypothetical operator splitting scheme, you will realize that any numerical method will only converge if you make the transport part converge. There are of course many ways of doing this, most notably things like SUPG or artificial diffusion. To understand why convergence for these methods does indeed happen, local truncation error analysis is not a very useful approach. You will need to read the literature on discretizations of transport equations and adapt the reasoning there.