An Accelerated Splitting-up Method for Parabolic Equations

Abstract

We approximate the solution u of the Cauchy problem \begin{gather*} \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, \end{gather*} by splitting the equation into the system $$ \frac{\partial}{\partial t} v_r(t,x)=L_rv_r(t,x)+f_r(t,x), \qquad r=1,2,\ldots,d_1, $$ 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 $W^m_p$, we show that for any $k>1$ one can approximate the solution u with an error of order $\delta^k$, by an appropriate combination of the solutions $v_r$ along a sequence of time discretization, where $\delta$ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [I. Gyöngy and N. Krylov, Ann. Probab., 31 (2003), pp. 564-591], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of $\delta$.