Semi-Lagrangian Advection Methods and Their Applications in Geoscience provides a much-needed resource on semi-Lagrangian theory, methods, and applications. Covering a variety of applications, the book brings together developments of the semi-Lagrangian in one place and offers a comparison of semi-Lagrangian methods with Eulerian-based approaches. It also includes a chapter dedicated to difficulties of dealing with the adjoint of semi-Lagrangian methods and illustrates the behavior of different schemes for different applications. This allows for a better understanding of which schemes are most efficient, stable, consistent, and likely to introduce the minimum model error into a given problem.
Beneficial for students learning about numerical approximations to advection, researchers applying these techniques to geoscientific modeling, and practitioners looking for the best approach for modeling, Semi-Lagrangian Advection Methods and Their Applications in Geoscience fills a crucial gap in numerical modeling and data assimilation in geoscience.
Author(s): Steven J. Fletcher
Edition: 1
Publisher: Elsevier
Year: 2019
Language: English
Pages: 615
Cover......Page 1
SEMI-LAGRANGIAN
ADVECTION
METHODS AND
THEIR
APPLICATIONS
IN GEOSCIENCE
......Page 4
Copyright
......Page 5
Contents......Page 6
1 Introduction......Page 9
2.1 Continuous form of the advection equation......Page 14
Mass conservation derivation of the advection equation......Page 15
2.1.2 Methods of characteristics......Page 18
2.2 Finite difference approximations to the Eulerian formulation of the advection equation......Page 21
2.2.1 Upwind forward Euler......Page 22
Upwind forward Euler with the bell curve......Page 24
Upwind forward Euler with the step function......Page 28
2.2.2 Forward-time, centered-space, FTCS......Page 30
2.2.3 Lax-Friedrichs scheme......Page 32
2.2.4 Lax-Wendroff scheme......Page 37
2.2.5 Leap-frog, centered-time-centered-space (CTCS)......Page 43
2.2.6 Linear multistep methods: Adams-Bashforth schemes......Page 46
2.2.7 Explicit Runge-Kutta methods......Page 51
Derivation of the general explicit fourth-order Runge-Kutta method......Page 52
2.3.1 Implicit, or backward Euler, scheme......Page 56
2.3.2 Crank-Nicolson scheme......Page 60
2.3.3 Box scheme......Page 61
2.3.4 Adams-Moulton methods......Page 62
2.3.5 Backward differentiation formula......Page 64
Implicit midpoint derivation......Page 66
Implicit trapezoidal derivation......Page 67
Collocated implicit Runge-Kutta schemes......Page 68
2.3.7 Diagonally implicit Runge-Kutta schemes (DIRK)......Page 71
2.4 Predictor-corrector methods......Page 72
2.5 Summary......Page 73
3.1 Truncation error......Page 75
3.1.1 Consistency......Page 78
3.1.2 Truncation errors and consistency analysis of the linear multistep methods......Page 79
3.2 Dispersion and dissipation errors......Page 83
3.3 Amplitude and phase errors......Page 84
3.4.1 Courant-Friedrichs-Lewy condition......Page 87
3.4.2 Von Neumann stability analysis......Page 88
3.4.3 Multistep method stability......Page 89
3.5 Quantifying the properties of the explicit finite difference schemes......Page 93
3.5.1 Upwind forward Euler scheme......Page 94
3.5.2 Forward-time-centered-space scheme......Page 97
3.5.3 Lax-Friedrichs scheme......Page 98
3.5.4 Lax-Wendroff scheme......Page 100
3.5.5 Leap-frog, centered-time-centered-space scheme......Page 101
3.6 Linear multistep methods......Page 102
3.6.1 Stability of Adams-Bashforth 2 scheme......Page 103
3.7 Consistency and stability of explicit Runge-Kutta methods......Page 106
3.8.1 Backward Euler scheme......Page 108
3.8.2 Crank-Nicolson scheme......Page 109
3.9 Predictor-corrector methods......Page 113
3.10 Summary......Page 114
4 History of semi-Lagrangian methods......Page 115
4.1.1 Barotropic problem......Page 117
4.1.2 The problem with time integration......Page 119
4.2 Welander (1955) paper......Page 121
4.3 Wiin-Nielsen (1959) paper......Page 126
4.4 Robert's (1981) paper......Page 130
4.5 Summary......Page 133
5.1 Derivation of the Lagrangian form for advection......Page 134
5.2 Derivation of the semi-Lagrangian approach......Page 136
5.3.1 Semi-Lagrangian advection using linear Lagrange interpolation......Page 140
5.3.2 Quadratic Lagrange interpolation polynomial......Page 142
5.3.3 Cubic Lagrange interpolation polynomial......Page 144
5.4 Semi-Lagrangian advection of the step function......Page 145
5.4.1 Linear Lagrange interpolation polynomial......Page 146
5.4.2 Quadratic Lagrange interpolation polynomial......Page 147
5.4.3 Cubic Lagrange interpolation polynomial......Page 149
5.5 Summary......Page 150
6 Interpolation methods......Page 152
6.1 Lagrange interpolation polynomials......Page 153
6.1.1 Illustrations of the Lagrange polynomials......Page 155
Continuous bell curve......Page 156
Discontinuous step function......Page 160
6.2 Newton divided difference interpolation polynomials......Page 163
6.3 Hermite interpolating polynomials......Page 168
6.4 Cubic spline interpolation polynomials......Page 171
6.5 Summary......Page 177
7.1 Stability of semi-Lagrangian schemes......Page 179
7.2.1 Linear Lagrange interpolation polynomial......Page 181
7.2.2 Stability analysis of the quadratic Lagrange interpolation......Page 184
7.2.3 Stability analysis of the cubic Lagrange interpolation......Page 188
7.3 Stability analysis of the cubic Hermite semi-Lagrangian interpolation scheme......Page 194
7.4 Stability analysis of the cubic spline semi-Lagrangian interpolation scheme......Page 199
7.5 Consistency analysis of semi-Lagrangian schemes......Page 203
7.6 Summary......Page 207
8.1 Semi-Lagrangian approaches for linear nonconstant advection velocity......Page 208
8.1.1 Numerical integration......Page 209
8.2 Two and three time level schemes......Page 212
Scheme (a)......Page 218
Scheme (b2)......Page 219
Scheme (c3)......Page 220
8.2.2 Construction of higher order schemes......Page 221
8.3 Semi-Lagrangian approximations to nonlinear advection......Page 222
Fixed point iteration methods......Page 224
Secant method......Page 226
Aliasing......Page 227
8.5 Nonlinear instability II......Page 229
8.5.1 Discrete approximations......Page 232
8.6 Boundary conditions for limit area models......Page 233
Time interpolation......Page 235
Well-posed buffer zone......Page 236
8.7 Summary......Page 237
9 Nonzero forcings......Page 238
9.1 Methods of characteristics approach......Page 239
9.1.1 Nonlinear shallow water equations......Page 242
9.2 Semi-implicit integration......Page 244
9.3 Semi-implicit semi-Lagrangian (SISL)......Page 246
9.4 Spatial averaging......Page 249
9.5 Optimal accuracy associated with uncentering time averages......Page 250
9.6 Semi-Lagrangian trajectories and discrete modes......Page 255
Linearization of the discrete equations......Page 257
9.7 Time-splitting......Page 261
9.8 Boundary conditions for the advection-adjustment equation......Page 264
Trajectory truncation......Page 266
Time interpolation......Page 267
Well-posed buffer zone......Page 268
9.9 Summary......Page 269
10.1 Bivariate interpolation methods......Page 271
10.1.1 Bilinear Lagrange interpolation......Page 273
10.1.2 Biquadratic Lagrange interpolation......Page 274
10.1.3 Bicubic Lagrange interpolation......Page 275
10.1.4 Biquartic Lagrange interpolation......Page 277
10.2.1 Arakawa A grid......Page 278
10.2.2 Arakawa B grid......Page 279
10.2.4 Arakawa D grid......Page 280
10.2.6 Vertical staggered grids......Page 281
10.3 Semi-implicit semi-Lagrangian finite differences in two dimensions......Page 282
10.3.1 Advection-diffusion in two dimensions......Page 283
10.4 Nonlinear shallow water equations......Page 286
10.5 Finite element based semi-Lagrangian method......Page 289
10.5.1 Weak Galerkin finite-element discretization......Page 291
10.6 Semi-Lagrangian integration in flux form......Page 296
10.6.1 Flux-form semi-Lagrangian (FFSL) advection......Page 300
Extension to 2D......Page 302
10.7.1 One-dimensional cell-integrated semi-Lagrangian (CISL) schemes......Page 304
10.7.2 Von Neumann stability analysis......Page 305
10.7.3 Two-dimensional finite volume schemes......Page 307
10.8 Semi-Lagrangian advection in flows with rotation and deformation......Page 309
10.8.1 Higher-order time accuracy......Page 311
10.9 Eliminating the interpolation......Page 313
10.9.1 Application in the shallow water equations model......Page 317
10.10 Semi-Lagrangian approach with ocean circulation models......Page 320
10.10.1 Implementing the semi-Lagrangian approach......Page 321
10.11 Transparent boundary conditions......Page 322
10.11.1 Waves at the boundaries......Page 325
10.11.2 Semi-Lagrangian discretization of transparent boundaries......Page 327
10.11.3 Characteristic boundaries......Page 328
10.11.4 Implementing semi-Lagrangian approach......Page 329
10.11.5 Discretization on the boundaries......Page 330
Gaussian bell......Page 333
Solid body rotation......Page 334
C0 cone......Page 335
Two deformational flows......Page 336
10.13 Semi-Lagrangian methods with the 2D quasi-geostrophic potential vorticity (Eady model)......Page 337
10.13.1 Buoyancy advection on the boundaries: b´ 0=0, b´1=αsin(KΔx)......Page 339
10.13.2 QGPV<>0......Page 342
10.14 Summary......Page 348
11.1 Trivariate interpolation methods......Page 353
11.2 Semi-Lagrangian advection in the primitive equations......Page 357
11.2.1 3D semi-Lagrangian integration......Page 361
11.3 3D flux form semi-Lagrangian......Page 362
11.4 Three-dimensional fully elastic Euler equations with semi-Lagrangian......Page 363
Vertical coordinate......Page 364
11.4.1 Time filters......Page 369
11.5 Sensitivity to departure point calculations......Page 370
11.6 Consistency of semi-Lagrangian trajectory calculations......Page 374
11.6.1 Trajectory calculations......Page 375
11.7 Semi-implicit Eulerian Lagrangian finite elements (SELFE)......Page 376
11.7.1 Numerical formulation of SELFE......Page 377
11.7.2 Semi-Lagrangian advection in SELFE......Page 381
11.8 Summary......Page 382
12.1.1 Spherical unit vectors......Page 383
12.1.2 Spherical vector derivative operators......Page 384
12.2 Grid development for a sphere......Page 385
Stereographic projection......Page 386
Mercator projection......Page 387
Lambert conic projection......Page 388
Sinusoidal projection......Page 389
12.3 Grid-point representations of the sphere......Page 390
12.3.1 Rectangular/square grids......Page 391
12.3.3 Hexagonal grid......Page 392
12.3.4 Cubed sphere......Page 395
12.4 Spectral modeling......Page 396
12.4.1 One-dimensional linear advection equation example......Page 399
12.4.2 Nonlinear advection (Burgers equation)......Page 401
12.4.4 Fast Fourier transforms......Page 402
12.4.5 Sturm-Liouville theory......Page 404
12.4.6 Legendre differential equation......Page 405
12.4.7 Legendre polynomials......Page 407
12.4.8 Spherical harmonics......Page 409
12.4.9 Legendre transforms......Page 411
12.4.10 Spectral methods on the sphere......Page 412
12.5 Semi-Lagrangian and alternating direction implicit (SLADI) scheme......Page 415
12.6 Global semi-Lagrangian modeling of the shallow water equations......Page 418
12.6.1 Ritchie (1988) approach......Page 419
12.6.2 Côte and Staniforth (1988) approach......Page 426
12.6.3 McDonald and Bates (1989) approach......Page 430
12.7 Spectral modeling of the shallow water equations......Page 437
12.7.1 Conformal transformation......Page 438
Stretching......Page 439
12.7.2 Semi-implicit semi-Lagrangian scheme in spectral space......Page 440
12.7.3 Treatment of Coriolis terms......Page 442
12.8 Semi-implicit semi-Lagrangian scheme on the sphere......Page 444
12.9 Removing the Helmholtz equation......Page 449
12.10 Stable extrapolation two-time-level scheme (SETTLS)......Page 451
12.10.1 SETTLS......Page 453
Stability analysis......Page 454
12.11 Flux form on a sphere......Page 458
12.11.1 Multidimensional flux-form semi-Lagrangian scheme......Page 459
12.11.2 2D FFSL with the shallow water equations......Page 460
12.11.3 Lagrangian control-volume......Page 465
12.12 Numerical test cases for the sphere......Page 466
Advection of cosine bell over the pole......Page 467
Global steady state nonlinear zonal geostrophic flow......Page 468
Rossby-Haurwitz wave......Page 469
12.13 Summary......Page 470
13.1 Shape-preserving semi-Lagrangian advection......Page 472
Monotonic shape-preserving constraints......Page 475
An aside to Fritsch and Carlson (1980) paper......Page 476
Returning to Williamson and Rasch (1989) and (1990) papers......Page 478
Fractional time steps......Page 480
Monotonic bicubic interpolation......Page 481
13.2 Cascade interpolation......Page 483
Lagrange interpolation......Page 484
Interpolation using hybrid grids......Page 485
13.2.2 Monotonic cascade interpolation......Page 487
13.2.3 Cascade interpolation on a sphere......Page 490
Determining of mesh intersection points......Page 492
Interpolating along the Lagrangian longitudes Θi......Page 493
13.3 Semi-Lagrangian inherently conserving and efficient scheme (SLICE)......Page 494
13.3.1 Piecewise cubic method......Page 496
13.3.2 SLICE 2D......Page 498
13.3.3 SLICE-S: SLICE on the sphere......Page 501
Special IECV (SIECVs)......Page 504
Correction of the polar cap region......Page 505
13.3.4 C-SLICE......Page 506
Simplified computation of intersections......Page 507
Simplified computation of distances......Page 508
Further aspects for spherical geometry......Page 509
13.3.5 SLICE applied to global SISL SWE model......Page 510
Spatial discretization......Page 513
Helmholtz equation......Page 514
13.3.6 SLICE-3D......Page 516
Grid definitions......Page 517
Vertical intersections......Page 518
13.3.7 Applications of SLICE-3D: non-hydrostatic vertical-slice equations......Page 519
13.4 Flux-form semi-Lagrangian spectral element approach......Page 522
Time-integrated formulation......Page 526
13.5 Conservative semi-Lagrangian HWENO method for the Vlasov equations......Page 528
13.5.1 HWENO reconstruction for flux functions......Page 531
13.5.2 Strang splitting SL HWENO scheme for the Vlasov-Poisson system......Page 534
13.6 Summary......Page 541
14.1 Derivation of the linearized model......Page 542
14.2 Adjoints......Page 543
14.3 Test of the tangent linear and adjoint models......Page 546
14.4 Differentiating the code to derive the adjoint......Page 547
14.5 Tangent linear approximations to semi-Lagrangian schemes......Page 549
14.5.1 Linearizing semi-Lagrangian interpolation......Page 551
Tanguay and Polavarapu (1999) paper......Page 554
Adjoints of cases A, B, and C......Page 557
14.5.2 Adjoint of 2D tracer on the sphere......Page 559
14.6 Perturbation forecast modeling......Page 560
14.6.1 Example with a 1D shallow water equations model......Page 561
Numerical models......Page 562
14.7 Sensitivity of adjoint of semi-Lagrangian integration to departure point iterations......Page 563
14.8 Summary......Page 567
15.1 Atmospheric sciences......Page 569
15.1.1 Moisture transport......Page 571
15.1.2 Convection scale......Page 574
15.2.1 Pollutant transport......Page 576
15.2.2 Volcanic ash transport......Page 577
15.3.2 Tidal flow simulations......Page 580
15.3.3 Displacement of free surface......Page 581
15.3.5 Coastal ocean models......Page 583
15.4 Earth's mantle and interior......Page 586
15.4.2 Crustal accretion......Page 587
15.4.3 Hydrothermal circulation......Page 588
15.5 Other applications......Page 591
15.6 Summary......Page 592
Solution to Exercise 2.2......Page 594
Solution to Exercise 6.4......Page 595
Bibliography......Page 596
Index......Page 604
Back Cover......Page 615
