An accelerated splitting-up method for parabolic equations

We approximate the solution u of the Cauchy problem ∂/∂t u(t, x) - Lu(t, x) + f(t, x), (t, x) ∈ (0, T] × ℝ ^d, u(0, x) =u ₀(x), x ∈ ℝ ^d, by splitting the equation into the system ∂/∂t ν _r(t, x) = L _rν _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 -∑ _r L _r, f =∑ _r fr. 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 appropriste combination of the solutions ν _r along a sequence of time discretisation, where δ 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. 564-691], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of δ.