Smoothing with positivity-preserving Padé schemes for parabolic problems with nonsmooth data

We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well-known Rannacher schemes. The new family of algorithms utilizes diagonal Fade schemes combined with positivity-preserving Fade schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self-adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics.