A generalization of Peaceman-Rachford fractional step method

In this paper we develop a set of time integrators of type fractional step Runge-Kutta methods which generalize the time integrator involved in the classical Peaceman-Rachford scheme. Combining a time semidiscretization of this type with a standard spatial discretization, we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered. We prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute. Finally, we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition.

[1]  J. C. Jorge,et al.  Special meshes and domain decomposition methods for evolutionary convection–diffusion–reaction problems , 2005 .

[2]  G. Marchuk.  Splitting and alternating direction methods , 1990 .

[3]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[4]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[5]  Isaías Alonso-Mallo,et al.  Spectral-fractional step Runge-Kutta discretizations for initial boundary value problems with time dependent boundary conditions , 2004, Math. Comput..

[6]  Junping Wang,et al.  Domain Decomposition Operator Splittings for the Solution of Parabolic Equations , 1998, SIAM J. Sci. Comput..

[7]  Laura Portero,et al.  A Combined Fractional Step Domain Decomposition Method for the Numerical Integration of Parabolic Problems , 2003, PPAM.

[8]  J. C. Jorge,et al.  Avoiding order reduction of fractional step Runge-Kutta discretizations for linear time dependent coefficient parabolic problems , 2004 .

[9]  Graeme Fairweather,et al.  A New Computational Procedure for A.D.I. Methods , 1967 .

[10]  J. M. Sanz-Serna,et al.  Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations , 1987 .

[11]  J. J. Douglas.  On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods , 1955 .

[12]  B. Bujanda,et al.  Fractional step Runge--Kutta methods for time dependent coefficient parabolic problems , 2003 .