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| When are High Order Methods Effective? | |
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| Preliminaries | |
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| Wave Propagation Problems | |
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| Parabolic Equations | |
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| Schrodinger Type Equations | |
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| Summary | |
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| Well-posedness and Stability | |
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| Well Posed Problems | |
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| Periodic Problems and Fourier Analysis | |
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| The PDE Problem | |
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| Difference Approximations | |
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| Initial-Boundary Value Problems and the Energy Method | |
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| The PDE Problem | |
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| Semidiscrete Approximations | |
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| Fully Discrete Approximations | |
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| Initial-Boundary Value Problems and Normal Mode Analysis for Hyperbolic Systems | |
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| Semidiscrete Approximations | |
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| Fully Discrete Approximations | |
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| Summary | |
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| Order of Accuracy and the Convergence Rate | |
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| Periodic Solutions | |
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| Initial-Boundary Value Problems | |
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| Summary | |
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| Approximation in Space | |
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| High Order Formulas on Standard Grids | |
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| High Order Formulas on Staggered Grids | |
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| Compact Pade Type Difference Operators | |
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| Optimized Difference Operators | |
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| Summary | |
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| Approximation in Time | |
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| Stability and the Test Equation | |
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| Runge-Kutta Methods | |
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| Linear Multistep Methods | |
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| Deferred Correction | |
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| Richardson Extrapolation | |
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| Summary | |
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| Coupled Space-Time Approximations | |
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| Taylor Expansions and the Lax-Wendroff Principle | |
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| Implicit Fourth Order Methods | |
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| Summary | |
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| Boundary Treatment | |
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| Numerical Boundary Conditions | |
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| Summation by Parts (SBP) Difference Operators | |
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| SBP Operators and Projection Methods | |
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| SBP Operators and Simultaneous Approximation Term (SAT) Methods | |
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| Summary | |
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| The Box Scheme | |
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| The Original Box Scheme | |
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| The Shifted Box Scheme | |
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| Two Space Dimensions | |
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| Nonuniform Grids | |
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| Summary | |
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| Wave Propagation | |
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| The Wave Equation | |
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| One Space Dimension | |
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| Two Space Dimensions | |
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| Discontinuous Coefficients | |
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| The Original One Step Scheme | |
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| Modified Coefficients | |
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| An Example with Discontinuous Solution | |
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| Boundary Treatment | |
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| High Order Boundary Conditions | |
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| Embedded Boundaries | |
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| Summary | |
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| A Problem in Fluid Dynamics | |
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| Large Scale Fluid Problems and Turbulent Flow | |
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| Stokes Equations for Incompressible Flow | |
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| A Fourth Order Method for Stokes Equations | |
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| Navier-Stokes Equations for Incompressible Flow | |
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| A Fourth Order Method for Navier-Stokes Equations | |
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| Summary | |
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| Nonlinear Problems with Shocks | |
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| Difference Methods and Nonlinear Equations | |
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| Conservation Laws | |
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| Shock Fitting | |
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| Artificial Viscosity | |
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| Upwind Methods | |
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| ENO and WENO Schemes | |
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| Summary | |
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| Introduction to Other Numerical Methods | |
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| Finite Element Methods | |
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| Galerkin FEM | |
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| Petrov-Galerkin FEM | |
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| Discontinuous Galerkin Methods | |
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| Spectral Methods | |
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| Fourier Methods | |
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| Polynomial Methods | |
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| Finite Volume Methods | |
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| Solution of Difference Equations | |
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| The Form of SBP Operators | |
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| Diagonal H-norm | |
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| Full H[subscript 0]-norm | |
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| A Pade Type Operator | |
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| References | |
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| Index | |