An accelerated splitting-up method for parabolic equations

István Gyöngy; Nicolai Krylov

paper

An accelerated splitting-up method for parabolic equations

arXiv:math/0412338

Abstract

We approximate the solution $u$ of the Cauchy problem $$ \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d, $$ $$ u(0,x)=u_0(x),\quad x\in\bR^d $$ by splitting the equation into the system $\frac{\partial}{\partial t} v_{r} (t, x) = L_{r} v_{r} (t, x) + f_{r} (t, x), r = 1, 2, ..., d_{1},$ where $L, L_{r}$ are second order differential operators, $f$ , $f_{r}$ are functions of $t, x$ , such that $L = \sum_{r} L_{r}$ , $f = \sum_{r} f_{r}$ . Under natural conditions on solvability in the Sobolev spaces $W_{p}^{m}$ , we show that for any $k > 1$ one can approximate the solution $u$ with an error of order $δ^{k}$ , by an appropriate combination of the solutions $v_{r}$ along a sequence of time discretization, where $δ$ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [7], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of $δ$ .

34 pages