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A note on local refinement for direction splitting methods

: Gornak, T.; Iliev, O.; Minev, P.


Dimov, I.:
Numerical methods and applications. 8th international conference, NMA 2014 : Borovets, Bulgaria, August 20 - 24, 2014; Revised selected papers
Cham: Springer International Publishing, 2015 (Lecture Notes in Computer Science 8962)
ISBN: 978-3-319-15584-5 (Print)
ISBN: 978-3-319-15585-2 (Online)
International Conference on Numerical Methods and Applications (NMA) <8, 2014, Borovets>
Conference Paper
Fraunhofer ITWM ()

In this note we propose a grid refinement procedure for direction splitting schemes for parabolic problems that can be easily extended to the incompressible Navier-Stokes equations. The procedure is developed to be used in conjunction with a direction splitting time discretization. Therefore, the structure of the resulting linear systems is tridiagonal for all internal unknowns, and only the Schur complement matrix for the unknowns at the interface of refinement has a four diagonal structure. Then the linear system in each direction can be solved either by a kind of domain decomposition iteration or by a direct solver, after an explicit computation of the Schur complement. The numerical results on a manufactured solution demonstrate that this grid refinement procedure does not alter the spatial accuracy of the finite difference approximation and seems to be unconditionally stable.