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Title: A B-Spline Method for Solving the Navier Stokes Equations

Abstract

Collocation methods using piece-wise polynomials, including B-splines, have been developed to find approximate solutions to both ordinary and partial differential equations. Such methods are elegant in their simplicity and efficient in their application. The spline collocation method is typically more efficient than traditional Galerkin finite element methods, which are used to solve the equations of fluid dynamics. The collocation method avoids integration. Exact formulae are available to find derivatives on spline curves and surfaces. The primary objective of the present work is to determine the requirements for the successful application of B-spline collocation to solve the coupled, steady, 2D, incompressible Navier–Stokes and continuity equations for laminar flow. The successful application of B-spline collocation included the development of ad hoc method dubbed the Boundary Residual method to deal with the presence of the pressure terms in the Navier–Stokes equations. Historically, other ad hoc methods have been developed to solve the incompressible Navier–Stokes equations, including the artificial compressibility, pressure correction and penalty methods. Convergence studies show that the ad hoc Boundary Residual method is convergent toward an exact (manufactured) solution for the 2D, steady, incompressible Navier–Stokes and continuity equations. C1 cubic and quartic B-spline schemes employing orthogonal collocation and C2 cubic andmore » C3 quartic B-spline schemes with collocation at the Greville points are investigated. The C3 quartic Greville scheme is shown to be the most efficient scheme for a given accuracy, even though the C1 quartic orthogonal scheme is the most accurate for a given partition. Two solution approaches are employed, including a globally-convergent zero-finding Newton's method using an LU decomposition direct solver and the variable-metric minimization method using BFGS update.« less

Authors:
Publication Date:
Research Org.:
Idaho National Lab. (INL), Idaho Falls, ID (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
912020
Report Number(s):
INEEL/JOU-02-00347
Journal ID: ISSN 0045-7930; CPFLBI; TRN: US0800265
DOE Contract Number:  
DE-AC07-99ID-13727
Resource Type:
Journal Article
Journal Name:
Computers and Fluids
Additional Journal Information:
Journal Volume: 34; Journal Issue: 1; Journal ID: ISSN 0045-7930
Country of Publication:
United States
Language:
English
Subject:
99 - GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ACCURACY; COMPRESSIBILITY; CONTINUITY EQUATIONS; CONVERGENCE; FINITE ELEMENT METHOD; LAMINAR FLOW; MINIMIZATION; NAVIER-STOKES EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; B-splines; equations; methods; Navier-Stokes

Citation Formats

Johnson, Richard Wayne. A B-Spline Method for Solving the Navier Stokes Equations. United States: N. p., 2005. Web. doi:10.1016/j.compfluid.2004.03.005.
Johnson, Richard Wayne. A B-Spline Method for Solving the Navier Stokes Equations. United States. https://doi.org/10.1016/j.compfluid.2004.03.005
Johnson, Richard Wayne. 2005. "A B-Spline Method for Solving the Navier Stokes Equations". United States. https://doi.org/10.1016/j.compfluid.2004.03.005.
@article{osti_912020,
title = {A B-Spline Method for Solving the Navier Stokes Equations},
author = {Johnson, Richard Wayne},
abstractNote = {Collocation methods using piece-wise polynomials, including B-splines, have been developed to find approximate solutions to both ordinary and partial differential equations. Such methods are elegant in their simplicity and efficient in their application. The spline collocation method is typically more efficient than traditional Galerkin finite element methods, which are used to solve the equations of fluid dynamics. The collocation method avoids integration. Exact formulae are available to find derivatives on spline curves and surfaces. The primary objective of the present work is to determine the requirements for the successful application of B-spline collocation to solve the coupled, steady, 2D, incompressible Navier–Stokes and continuity equations for laminar flow. The successful application of B-spline collocation included the development of ad hoc method dubbed the Boundary Residual method to deal with the presence of the pressure terms in the Navier–Stokes equations. Historically, other ad hoc methods have been developed to solve the incompressible Navier–Stokes equations, including the artificial compressibility, pressure correction and penalty methods. Convergence studies show that the ad hoc Boundary Residual method is convergent toward an exact (manufactured) solution for the 2D, steady, incompressible Navier–Stokes and continuity equations. C1 cubic and quartic B-spline schemes employing orthogonal collocation and C2 cubic and C3 quartic B-spline schemes with collocation at the Greville points are investigated. The C3 quartic Greville scheme is shown to be the most efficient scheme for a given accuracy, even though the C1 quartic orthogonal scheme is the most accurate for a given partition. Two solution approaches are employed, including a globally-convergent zero-finding Newton's method using an LU decomposition direct solver and the variable-metric minimization method using BFGS update.},
doi = {10.1016/j.compfluid.2004.03.005},
url = {https://www.osti.gov/biblio/912020}, journal = {Computers and Fluids},
issn = {0045-7930},
number = 1,
volume = 34,
place = {United States},
year = {Sat Jan 01 00:00:00 EST 2005},
month = {Sat Jan 01 00:00:00 EST 2005}
}