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.
- Cauchy problem
- Method of alternative direction
- Parabolic partial differential equations
- Richardson's method