| When are High Order Methods Effective? | p. 1 |
| Preliminaries | p. 1 |
| Wave Propagation Problems | p. 2 |
| Parabolic Equations | p. 8 |
| Schrodinger Type Equations | p. 11 |
| Summary | p. 12 |
| Well-posedness and Stability | p. 13 |
| Well Posed Problems | p. 13 |
| Periodic Problems and Fourier Analysis | p. 16 |
| The PDE Problem | p. 17 |
| Difference Approximations | p. 21 |
| Initial-Boundary Value Problems and the Energy Method | p. 29 |
| The PDE Problem | p. 29 |
| Semidiscrete Approximations | p. 33 |
| Fully Discrete Approximations | p. 38 |
| Initial-Boundary Value Problems and Normal Mode Analysis for Hyperbolic Systems | p. 41 |
| Semidiscrete Approximations | p. 41 |
| Fully Discrete Approximations | p. 59 |
| Summary | p. 66 |
| Order of Accuracy and the Convergence Rate | p. 69 |
| Periodic Solutions | p. 69 |
| Initial-Boundary Value Problems | p. 72 |
| Summary | p. 79 |
| Approximation in Space | p. 81 |
| High Order Formulas on Standard Grids | p. 81 |
| High Order Formulas on Staggered Grids | p. 85 |
| Compact Pade Type Difference Operators | p. 87 |
| Optimized Difference Operators | p. 91 |
| Summary | p. 93 |
| Approximation in Time | p. 95 |
| Stability and the Test Equation | p. 95 |
| Runge-Kutta Methods | p. 97 |
| Linear Multistep Methods | p. 102 |
| Deferred Correction | p. 108 |
| Richardson Extrapolation | p. 111 |
| Summary | p. 113 |
| Coupled Space-Time Approximations | p. 115 |
| Taylor Expansions and the Lax-Wendroff Principle | p. 115 |
| Implicit Fourth Order Methods | p. 117 |
| Summary | p. 124 |
| Boundary Treatment | p. 127 |
| Numerical Boundary Conditions | p. 127 |
| Summation by Parts (SBP) Difference Operators | p. 130 |
| SBP Operators and Projection Methods | p. 140 |
| SBP Operators and Simultaneous Approximation Term (SAT) Methods | p. 147 |
| Summary | p. 155 |
| The Box Scheme | p. 157 |
| The Original Box Scheme | p. 157 |
| The Shifted Box Scheme | p. 161 |
| Two Space Dimensions | p. 165 |
| Nonuniform Grids | p. 169 |
| Summary | p. 176 |
| Wave Propagation | p. 177 |
| The Wave Equation | p. 177 |
| One Space Dimension | p. 178 |
| Two Space Dimensions | p. 185 |
| Discontinuous Coefficients | p. 192 |
| The Original One Step Scheme | p. 193 |
| Modified Coefficients | p. 201 |
| An Example with Discontinuous Solution | p. 206 |
| Boundary Treatment | p. 209 |
| High Order Boundary Conditions | p. 209 |
| Embedded Boundaries | p. 210 |
| Summary | p. 216 |
| A Problem in Fluid Dynamics | p. 219 |
| Large Scale Fluid Problems and Turbulent Flow | p. 219 |
| Stokes Equations for Incompressible Flow | p. 220 |
| A Fourth Order Method for Stokes Equations | p. 223 |
| Navier-Stokes Equations for Incompressible Flow | p. 228 |
| A Fourth Order Method for Navier-Stokes Equations | p. 231 |
| Summary | p. 242 |
| Nonlinear Problems with Shocks | p. 245 |
| Difference Methods and Nonlinear Equations | p. 245 |
| Conservation Laws | p. 246 |
| Shock Fitting | p. 251 |
| Artificial Viscosity | p. 252 |
| Upwind Methods | p. 257 |
| ENO and WENO Schemes | p. 261 |
| Summary | p. 265 |
| Introduction to Other Numerical Methods | p. 267 |
| Finite Element Methods | p. 267 |
| Galerkin FEM | p. 267 |
| Petrov-Galerkin FEM | p. 281 |
| Discontinuous Galerkin Methods | p. 283 |
| Spectral Methods | p. 289 |
| Fourier Methods | p. 290 |
| Polynomial Methods | p. 295 |
| Finite Volume Methods | p. 300 |
| Solution of Difference Equations | p. 307 |
| The Form of SBP Operators | p. 311 |
| Diagonal H-norm | p. 311 |
| Full H[subscript 0]-norm | p. 317 |
| A Pade Type Operator | p. 323 |
| References | p. 325 |
| Index | p. 331 |
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