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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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Finite Difference Methods

Finite Difference Methods

(p.186) 14 Finite Difference Methods
Fluid Mechanics

S. G. Rajeev

Oxford University Press

This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.

Keywords:   Finite difference method, Padé approximant, Poisson equation, di?usion equation, Clebsch formulation, radial basis function

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