Numerical Analysis and Scientific Computing Seminar | Christina C. Christara, Convergence of high-order methods on parabolic PDEs with nonsmooth initial conditions

Several PDE problems, including pricing of financial contracts, involve initial conditions that exhibit various types of nonsmoothness at certain points of the spatial domain, usually known in advance. Such nonsmoothness not only renders the standard error analysis inapplicable (or not directly applicable), but results in erratic error convergence of computational methods. This is noticed in standard second-order methods, but is even more prominent in high-order methods.

We present an analysis of the error arising from certain types of nonsmooth initial conditions in the numerical solution of a parabolic PDE, focusing on high-order methods. While the methods considered are fourth-order finite differences in space, and BDF4 timestepping with initialization by 2 RK3 and 1 BDF3 steps, the analysis gives insight to general treatment of various sources of errors, such as the so-called quantization error and the spurious oscillations. Traditional (though not as well-known) and novel smoothing techniques are introduced to restore and stabilize the high order of convergence. Numerical examples on the model PDE and various option pricing problems are also given to demonstrate the fourth-order convergence of our method.