Approximate solutions to second-order parabolic equations: Evolution systems and discretization
Wen Cheng, Equity Derivative Quantitative Strategies, Credit Suisse, 11 Madison Avenue, New York, NY 10010, USA, Anna L. Mazzucato, Victor Nistor, Université' de Lorraine, 3, rue Augustin Fresnel, 57000 Metz, France · Discrete and Continuous Dynamical Systems - S · 2022
We study the discretization of a linear evolution partial differential equation when its Green's function is known or well approximated. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method that we developed in J. Math. Phys. 51 (2010), n. 10, 103502 (reference [ 15 ]). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.