Contents
xv
4 Series-Expansion Methods
173
4.1 Strategies for Minimizing the Residual. . . . . . . .
173
4.2 The Spectral Method . . . . . . . . . . . . . . . . .
176
4.2.1 Comparison with Finite-Difference Methods .
4.2.2 Improving Efficiency Using the Transform Method
4.2.3 Conservation and the Galerkin Approximation .
4.3 The Pseudospectral Method . . .
177
184
189
191
4.4 Spherical Harmonics . . . . . . . . . .
195
4.4.1 Truncating the Expansion . . .
197
4.4.2 Elimination of the Pole Problem
200
4.4.3 Gaussian Quadrature and the Transform Method .
202
4.4.4 Nonlinear Shallow-Water Equations . . . . . . .
207
4.5 The Finite-Element Method . . . . . . . . . . . . . . .
212
4.5.1 Galerkin Approximation with Chapeau Functions
214
4.5.2 Petrov-Galerkin and Taylor-Galerkin Methods
216
4.5.3 Quadratic Expansion Functions .. ..
219
4.5.4 Hermite-Cubic Expansion Functions . .
226
4.5.5 Two-Dimensional Expansion Functions
231
Problems . . . . . . . . . . . . . . . . . . . . . . .
234
5 Finite Volume Methods
241
5.1 Conservation Laws and Weak Solutions
5.1.1 The Riemann Problem . . . . .
243
244
5.1.2 Entropy-Consistent Solutions .
246
5.2 Finite-Volume Methods and Convergence
249
5.2.1 Monotone Schemes
251
5.2.2 TVD Methods . . . . . . . . . .
252
5.3 Discontinuities in Geophysical Fluid Dynamics
254
5. 5.4.1 4.1 F F1 1u ux x Corr Correctio ection n: : T The he Ori Original ginal Propo Proposal sal
25 259 9
5. 5.4.2 4.2 The The Zale Zales sak ak C Co or rre recto ctor r . .
26 260 0
5. 5.4.3 4.3 Iterative Iterative F F1 1u ux x Corr Correcti ection on . . . . . . . . . .
26 263 3
5.5 Flux-Limiter Methods . . . . . . . . . .
263
5.5.1 Ensuring That the Scheme Is TVD
264
5.5.2 Possible F1ux Limiters . . . . . .
267
5.6
5.7
5.5.3 F10w Velocities of Arbitrary Sign
Approximation with Local Polynomials
5.6.1 Godunov's Method . . . . .
5.6.2 Piecewise-Linear Functions
Two Spatial Dimensions . . . . . .
271
272
272
274
277
5.7.1
5.7.2
5.7.3
5.7.4
5.7.5
FCT in Two Dimensions . .
F1ux-Limiter Methods for Uniform 2-D F10w
Nonuniform Nondivergent F10w . .
A Numerical Example . . . . . . .
When Is a F1ux Limiter Necessary?
277
279
282
284
291
xv
4 Series-Expansion Methods
173
4.1 Strategies for Minimizing the Residual. . . . . . . .
173
4.2 The Spectral Method . . . . . . . . . . . . . . . . .
176
4.2.1 Comparison with Finite-Difference Methods .
4.2.2 Improving Efficiency Using the Transform Method
4.2.3 Conservation and the Galerkin Approximation .
4.3 The Pseudospectral Method . . .
177
184
189
191
4.4 Spherical Harmonics . . . . . . . . . .
195
4.4.1 Truncating the Expansion . . .
197
4.4.2 Elimination of the Pole Problem
200
4.4.3 Gaussian Quadrature and the Transform Method .
202
4.4.4 Nonlinear Shallow-Water Equations . . . . . . .
207
4.5 The Finite-Element Method . . . . . . . . . . . . . . .
212
4.5.1 Galerkin Approximation with Chapeau Functions
214
4.5.2 Petrov-Galerkin and Taylor-Galerkin Methods
216
4.5.3 Quadratic Expansion Functions .. ..
219
4.5.4 Hermite-Cubic Expansion Functions . .
226
4.5.5 Two-Dimensional Expansion Functions
231
Problems . . . . . . . . . . . . . . . . . . . . . . .
234
5 Finite Volume Methods
241
5.1 Conservation Laws and Weak Solutions
5.1.1 The Riemann Problem . . . . .
243
244
5.1.2 Entropy-Consistent Solutions .
246
5.2 Finite-Volume Methods and Convergence
249
5.2.1 Monotone Schemes
251
5.2.2 TVD Methods . . . . . . . . . .
252
5.3 Discontinuities in Geophysical Fluid Dynamics
254
5. 5.4.1 4.1 F F1 1u ux x Corr Correctio ection n: : T The he Ori Original ginal Propo Proposal sal
25 259 9
5. 5.4.2 4.2 The The Zale Zales sak ak C Co or rre recto ctor r . .
26 260 0
5. 5.4.3 4.3 Iterative Iterative F F1 1u ux x Corr Correcti ection on . . . . . . . . . .
26 263 3
5.5 Flux-Limiter Methods . . . . . . . . . .
263
5.5.1 Ensuring That the Scheme Is TVD
264
5.5.2 Possible F1ux Limiters . . . . . .
267
5.6
5.7
5.5.3 F10w Velocities of Arbitrary Sign
Approximation with Local Polynomials
5.6.1 Godunov's Method . . . . .
5.6.2 Piecewise-Linear Functions
Two Spatial Dimensions . . . . . .
271
272
272
274
277
5.7.1
5.7.2
5.7.3
5.7.4
5.7.5
FCT in Two Dimensions . .
F1ux-Limiter Methods for Uniform 2-D F10w
Nonuniform Nondivergent F10w . .
A Numerical Example . . . . . . .
When Is a F1ux Limiter Necessary?
277
279
282
284
291