## Abstract

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_{r}ν _{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.

Original language | English (US) |
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Pages (from-to) | 1070-1097 |

Number of pages | 28 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - 2005 |

## Keywords

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