1. Traditional Galerkin Methods.- 1.1 Introduction.- 1.2 Simple Examples.- 1.2.1 An ordinary differential equation.- 1.2.2 An eigenvalue problem.- 1.2.3 Viscous flow in a channel.- 1.2.4 Unsteady heat conduction.- 1.2.5 Burgers’ equation.- 1.3 Method of Weighted Residuals.- 1.3.1 Subdomain method.- 1.3.2 Collocation method.- 1.3.3 Least-squares method.- 1.3.4 Method of moments.- 1.3.5 Galerkin method.- 1.3.6 Generalized Galerkin method.- 1.3.7 Comparison of weighted-residual methods.- 1.4 Connection with Other Methods.- 1.5 Theoretical Properties.- 1.6 Applications.- 1.6.1 Natural convection in a rectangular slot.- 1.6.2 Hydrodynamic stability.- 1.6.3 Acoustic transmission in ducts.- 1.6.4 Flow around inclined airfoils.- 1.6.5 Microstrip disc problem.- 1.6.6 Other applications of traditional Galerkin methods.- 1.7 Closure.- References.- 2. Computational Galerkin Methods.- 2.1 Limitations of the Traditional Galerkin Method.- 2.2 Solution for Nodal Unknowns.- 2.3 Use of Low-order Test and Trial Functions.- 2.4 Use of Finite Elements to Handle Complex Geometry.- 2.5 Use of Orthogonal Test and Trial Functions.- 2.6 Evaluation of Nonlinear Terms in Physical Space.- 2.7 Advantages of Computational Galerkin Methods.- 2.8 Closure.- References.- 3. Galerkin Finite-Element Methods.- 3.1 Trial Functions and Finite Elements.- 3.1.1 One-dimensional elements.- 3.1.2 Rectangular elements in two and three dimensions.- 3.1.3 Triangular elements.- 3.2 Examples.- 3.2.1 A simplified Sturm-Liouville equation.- 3.2.2 Viscous flow in a channel.- 3.2.3 Inviscid, incompressible flow.- 3.2.4 Unsteady heat conduction.- 3.2.5 Burgers’ equation.- 3.3 Connection with Finite-Difference Formulae.- 3.4 Theoretical Properties.- 3.4.1 Convergence.- 3.4.2 Error estimates.- 3.4.3 Optimal error estimates and superconvergence.- 3.4.4 Numerical convergence results.- 3.5 Applications.- 3.5.1 Convective heat transfer.- 3.5.2 Viscous incompressible flow.- 3.5.3 Jet-flap flows.- 3.5.4 Acoustic transmission in ducts.- 3.5.5 Tidal flows.- 3.5.6 Weather forecasting.- 3.6 Closure.- References.- 4. Advanced Galerkin Finite-Element Techniques.- 4.1 Time Splitting.- 4.1.1 Thermal entry problem.- 4.1.2 Viscous compressible flow.- 4.2 Least-squares Residual Fitting.- 4.3 Special Trial Functions.- 4.3.1 Singularities.- 4.3.2 Near-wall turbulent flows.- 4.3.3 Dorodnitsyn boundary-layer formulation.- 4.4 Integral Equations.- 4.4.1 Boundary-element method.- 4.5 Closure.- References.- 5. Spectral Methods.- 5.1 Choice of Trial Functions.- 5.2 Examples.- 5.2.1 Unsteady heat conduction.- 5.2.2 Burgers’ equation.- 5.3 Techniques for Improved Efficiency.- 5.3.1 Recurrence relations.- 5.3.2 Nonlinear terms.- 5.3.3 Time differencing.- 5.4 Alternative Spectral Methods.- 5.4.1 Pseudo spectral method.- 5.4.2 The tau method.- 5.5 Orthonormal Method of Integral Relations.- 5.6 Applications.- 5.6.1 Global atmospheric modeling.- 5.6.2 Direct turbulence simulation.- 5.6.3 Other spectral applications.- 5.7 Closure.- References.- 6. Comparison of Finite Difference, Finite Element and Spectral Methods.- 6.1 Problems and Partial Differential Equations.- 6.2 Boundary Conditions and Complex Boundary Geometry.- 6.3 Computational Efficiency.- 6.4 Ease of Coding and Flexibility.- 6.5 Test Cases.- 6.5.1 Burgers’ equation.- 6.5.2 Model parabolic equations.- 6.5.3 Passive scalar convection.- 6.5.4 Open ocean modeling.- 6.6 Closure.- References.- 7. Generalized Galerkin Methods.- 7.1 A Motivation.- 7.2 Theoretical Background.- 7.2.1 Petrov-Galerkin formulation.- 7.2.2 Construction of the test function, ?k.- 7.3 Steady Convection-Diffusion Problems.- 7.3.1 Higher-order one-dimensional formulations.- 7.3.2 Two-dimensional formulations.- 7.4 Parabolic Problems.- 7.4.1 Transient convection-diffusion equation.- 7.4.2 Burgers’ equation.- 7.5 Hyperbolic Problems.- 7.6 Closure.- References.- Appendix 1 Program BURG1.- Appendix 2 Program BURG4.
