xvi
Contents
5.8 Schemes for Positive Definite Advection .
292
5.8.1 An FCT Approach . . . . . . . .
293
5.8.2 Antidiffusion via Upstream Differencing .
294
5.9 Curvilinear Coordinates
296
Problems . . . . . . . . . .
297
6. 6.1 1 The The Scalar Scalar Advectio Advection n E Eq qu uation ation
30 305 5
6 6.1 .1.1 .1 C Constan onstant t V Ve el lo oc ci it ty y . . . .. .
30 305 5
6 6.1. .1.2 2 V Va ar ri ia ab bl le e V Ve el lo oc ci it ty y . . . . . . . .
31 310 0
6. 6.2 2 Forc Forcing ing in in the the L Lagrangian agrangian Frame Frame . .
31 313 3
6. 6.3 3 Sy Systems stems of of Equa Equatio tion ns s
31 318 8
6 6.3. .3.1 1 C Co omparis mpariso on n w with ith the the M Me ethod thod o of f Chara Charac cte teristics ristics
3 318 18
6 6.3. .3.2 2 Sem Semi- i-im implic plicit it S Sem emi i- -L Lagr agra angian ngian S Sc che hemes mes
32 320 0
6. 6.4 4 A Alte lter rn na ative tive T Tr ra ajec jecto tor ries ies
32 324 4
6. 6.4 4.1 .1 A A N Noninte oninterpolat rpolatin ing g L Leap eapfr frog og Sch Sche eme me . . . .
32 325 5
6. 6.4 4.2 .2 Inte Inter rpolat polatio ion n v via ia Para Param me etrized trized Ad Advection vection
32 327 7
6.5 Eulerian or Semi-Lagrangian?
330
Problems . . . . . . . . . . . . . . .
331
7 Physically Insignificant Fast Waves
335
7.1 The Projection Method . . . . . . . . . .
336
7.1.1 Forward-in-Time Implementation
337
7.1.2 Leapfrog Implementation . . . .
339
7.1.3 Solving the Poisson Equation for Pressure
340
7.2 The Semi-implicit Method . . . . . . . . . .
342
7.2.1 Large Time Steps and Poor Accuracy . .
343
7.2.2 A Prototype Problem. . . . . . . . . . .
345
7.2.3 Semi-implicit Solution of the Shallow-Water Equations
347
7.3
7.2.4 Semi-implicit Solution of the Euler Equations
7.2.5 Numerical Implementation .
Fractional-Step Methods . . . . . .
7.3.1 Complete Operator Splitting
7.3.2 Partially Split Operators . .
350
356
359
359
365
7.4 Summary of Schemes for Nonhydrostatic Models
371
7.5 The Hydrostatic Approximation . .
372
7.6 Primitive Equation Models . . . . . . . . . . . .
374
7.6.1 Pressure and a Coordinates
375
7.6.2 Spectral Representation of the Horizontal Structure
379
7.6.3 Vertical Differencing . . . . . . .
381
7.6.4 Energy Conservation . . . . . . .
383
7.6.5 Semi-implicit Time-Differencing .
387
Problems . . . . . . . . . . . . . . . . . . . .
389
Contents
5.8 Schemes for Positive Definite Advection .
292
5.8.1 An FCT Approach . . . . . . . .
293
5.8.2 Antidiffusion via Upstream Differencing .
294
5.9 Curvilinear Coordinates
296
Problems . . . . . . . . . .
297
6. 6.1 1 The The Scalar Scalar Advectio Advection n E Eq qu uation ation
30 305 5
6 6.1 .1.1 .1 C Constan onstant t V Ve el lo oc ci it ty y . . . .. .
30 305 5
6 6.1. .1.2 2 V Va ar ri ia ab bl le e V Ve el lo oc ci it ty y . . . . . . . .
31 310 0
6. 6.2 2 Forc Forcing ing in in the the L Lagrangian agrangian Frame Frame . .
31 313 3
6. 6.3 3 Sy Systems stems of of Equa Equatio tion ns s
31 318 8
6 6.3. .3.1 1 C Co omparis mpariso on n w with ith the the M Me ethod thod o of f Chara Charac cte teristics ristics
3 318 18
6 6.3. .3.2 2 Sem Semi- i-im implic plicit it S Sem emi i- -L Lagr agra angian ngian S Sc che hemes mes
32 320 0
6. 6.4 4 A Alte lter rn na ative tive T Tr ra ajec jecto tor ries ies
32 324 4
6. 6.4 4.1 .1 A A N Noninte oninterpolat rpolatin ing g L Leap eapfr frog og Sch Sche eme me . . . .
32 325 5
6. 6.4 4.2 .2 Inte Inter rpolat polatio ion n v via ia Para Param me etrized trized Ad Advection vection
32 327 7
6.5 Eulerian or Semi-Lagrangian?
330
Problems . . . . . . . . . . . . . . .
331
7 Physically Insignificant Fast Waves
335
7.1 The Projection Method . . . . . . . . . .
336
7.1.1 Forward-in-Time Implementation
337
7.1.2 Leapfrog Implementation . . . .
339
7.1.3 Solving the Poisson Equation for Pressure
340
7.2 The Semi-implicit Method . . . . . . . . . .
342
7.2.1 Large Time Steps and Poor Accuracy . .
343
7.2.2 A Prototype Problem. . . . . . . . . . .
345
7.2.3 Semi-implicit Solution of the Shallow-Water Equations
347
7.3
7.2.4 Semi-implicit Solution of the Euler Equations
7.2.5 Numerical Implementation .
Fractional-Step Methods . . . . . .
7.3.1 Complete Operator Splitting
7.3.2 Partially Split Operators . .
350
356
359
359
365
7.4 Summary of Schemes for Nonhydrostatic Models
371
7.5 The Hydrostatic Approximation . .
372
7.6 Primitive Equation Models . . . . . . . . . . . .
374
7.6.1 Pressure and a Coordinates
375
7.6.2 Spectral Representation of the Horizontal Structure
379
7.6.3 Vertical Differencing . . . . . . .
381
7.6.4 Energy Conservation . . . . . . .
383
7.6.5 Semi-implicit Time-Differencing .
387
Problems . . . . . . . . . . . . . . . . . . . .
389