Book contents
- Frontmatter
- Contents
- Preface
- PART ONE PRELIMINARIES
- PART TWO FINITE DIFFERENCE METHODS
- PART THREE FINITE ELEMENT METHODS
- 8 Introduction to Finite Element Methods
- 9 Finite Element Interpolation Functions
- 10 Linear Problems
- 11 Nonlinear Problems/Convection-Dominated Flows
- 12 Incompressible Viscous Flows via Finite Element Methods
- 13 Compressible Flows via Finite Element Methods
- 14 Miscellaneous Weighted Residual Methods
- 15 Finite Volume Methods via Finite Element Methods
- 16 Relationships between Finite Differences and Finite Elements and Other Methods
- PART FOUR FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES
- PART FIVE APPLICATIONS
- APPENDIXES
- Index
11 - Nonlinear Problems/Convection-Dominated Flows
Published online by Cambridge University Press: 15 January 2010
- Frontmatter
- Contents
- Preface
- PART ONE PRELIMINARIES
- PART TWO FINITE DIFFERENCE METHODS
- PART THREE FINITE ELEMENT METHODS
- 8 Introduction to Finite Element Methods
- 9 Finite Element Interpolation Functions
- 10 Linear Problems
- 11 Nonlinear Problems/Convection-Dominated Flows
- 12 Incompressible Viscous Flows via Finite Element Methods
- 13 Compressible Flows via Finite Element Methods
- 14 Miscellaneous Weighted Residual Methods
- 15 Finite Volume Methods via Finite Element Methods
- 16 Relationships between Finite Differences and Finite Elements and Other Methods
- PART FOUR FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES
- PART FIVE APPLICATIONS
- APPENDIXES
- Index
Summary
For fluid dynamics associated with nonlinearity and discontinuity, there have been significant developments in the last two decades both in finite difference methods (FDM) and finite element methods (FEM).Concurrent with upwind schemes in space and Taylor series expansion of variables in time for FDM formulations with various orders of accuracy, numerous achievements have been made in FEM applications since the publication of an earlier text [Chung, 1978].T hese new developments include generalized Galerkin methods (GGM), Taylor-Galerkin methods (TGM) [Donea, 1984], and the streamline upwind Petrov-Galerkin (SUPG) methods [Heinrich et al., 1977; Hughes and Brooks, 1982], alternatively referred to as the streamline diffusion method (SDM) [Johnson, 1987], and Galerkin/least squares (GLS) methods [Hughes and his
co-workers, 1988–1998].In the sections that follow, it will be shown that computational strategies such as SUPG or SDM and other similar methods can be grouped under the heading of generalized Petrov-Galerkin (GPG) methods.Recent developments include unstructured adaptive methods [Oden et al., 1986; Löhner, Morgan, and Zienkiewicz, 1985], characteristic Galerkin methods (CGM) [Zienkiewicz and his co-workers, 1994–1998], discontinuous Galerkin methods (DGM) [Oden and his co-workers, 1996–1998], and flowfield-dependent variation (FDV) methods [Chung and his coworkers, 1995–1999], among others.On the other hand, the concepts of FDM and FEM have been utilized in developing finite volume methods in conjunction with unstructured grids [Jameson, Baker, and Weatherill, 1986].It appears that FDM and FEM continue to co-exist and develop into a mature technology, mutually benefitting from each other.
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- Information
- Computational Fluid Dynamics , pp. 347 - 398Publisher: Cambridge University PressPrint publication year: 2002