Approximate solutions to second-order parabolic equations: Evolution systems and discretization

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 spatial approximation and time stepping approximation. show that, in fact, an Green almost as good itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations using Dyson-Taylor commutator method that developed J. Math. Phys. 51 (2010), n. 10, 103502 (reference [15]). This short time, combined with bootstrap argument, gives approximate solution on any fixed interval within prescribed tolerance.