Abstract
A new fourth order box-scheme for the Poisson problem in a square with Dirichlet boundary conditions is introduced, extending the approach in Croisille (Computing 78:329---353, 2006). The design is based on a box approach, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The goal is twofold: first to show that fourth order accuracy is obtained both for the unknown and the gradient; second, to describe a fast direct algorithm, based on the Sherman-Morrison formula and the Fast Sine Transform. Several numerical results in a square are given, indicating an asymptotic O(N 2log?2(N)) computing complexity.
Highlights
The design of high order compact finite-difference schemes for the Laplace equation in a squared or cubic geometry is a classical topic in Applied Mathematics and Scientific Computing
Beyond the design of specific numerical schemes, which deals with accuracy and stability, the need of an efficient fast solver is a crucial issue to perform practical computations. On this question we refer to the recent review [5] and the references therein. The use of such solvers in canonical geometries persists to be at the heart of many computing codes in physics
We introduce a new fourth order compact scheme on a cartesian grid for the Poisson problem in a rectangle, whose design is based on the preliminary work [17]
Summary
We refer to [13, 14] for examples of the numerous articles published in the 1960’s on the subject. Beyond the design of specific numerical schemes, which deals with accuracy and stability, the need of an efficient fast solver is a crucial issue to perform practical computations. On this question we refer to the recent review [5] and the references therein. The use of such solvers in canonical geometries persists to be at the heart of many computing codes in physics. Examples are fluid dymanics (compressible or incompressible Navier-Stokes equations), [34, 21, 3], the Helmholtz equation [9, 12], computations in astrophysics, [32] or in geophysics, [38]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
