vi CONTENTS 16 Coordinate Transformations 323 16.1 ntroduction...····· 323 l6.2 Programming Chebyshev Methods.·.··.·················· 323 16.3 Theory of 1-D Transformations.......................... 325 16.4 Infinite and Semi-Infinite Intervals 326 16.5 Maps for Endpoint Corner Singularities.. 327 16.6 Two-Dimensional Maps Corner Branch Points 329 l6.7 Periodic Problems&the Arctan./Tan Map..·.·...·.·.········ 330 16.8 Adaptive Methods 332 l6.9 Almost-Equispaced Kosloff/Tal-Ezer Grid...·.··...·...···.·. 334 17 Methods for Unbounded Intervals 338 17.1 ntroduction..··:··························· 338 l7.2 Domain Truncation..........·..。·...·.·······.····… 339 17.2.1 Domain Truncation for Rapidly-decaying Functions 339 17.2.2 Domain Truncation for Slowly-Decaying Functions 340 17.2.3 Domain Truncation for Time-Dependent Wave Propagation: Sponge Layers 340 17.3 Whittaker Cardinal or "Sinc"Functions 341 17.4 Hermite functions............... 346 17.5 Semi-Infinite Interval:Laguerre Functions... 353 l7.6 New Basis Sets via Change of Coordinate··.············· 355 l7.7 Rational Chebyshev Functions:TBn·.······.···. 356 17.8 Behavioral versus Numerical Boundary Conditions 361 17.9 Strategy for Slowly Decaying Functions.··········· 363 17.10Numerical Examples:Rational Chebyshev Functions 366 17.11Semi-Infinite Interval:Rational Chebyshev TLn 369 l7.l2 Numerical Examples:Chebyshev for Semi-Infinite Interval.....···.· 370 17.l3 Strategy:Oscillatory,Non-Decaying Functions············- 372 l7.l4 Weideman-Cloot Sinh Mapping.·,··.············· 374 17.15 Summary...·············· 377 18 Spherical Cylindrical Geometry 380 18.1 Introduction............... 380 18.2 Polar,Cylindrical,Toroidal,Spherical.... 381 18.3 Apparent Singularity at the Pole . 382 l8.4 Polar Coordinates:Parity Theorem.··········· 383 l8.5 Radial Basis Sets and Radial Grids........·...·.....·.···· 385 18.5.1 One-Sided Jacobi Basis for the Radial Coordinate 387 18.5.2 Boundary Value Eigenvalue Problems on a Disk 389 18.5.3 Unbounded Domains Including the Origin in Cylindrical Coordinates 390 18.6 Annular Domains..... 390 18.7 Spherical Coordinates:An Overview....................... 391 18.8 The Parity Factor for Scalars:Sphere versus Torus .............. 391 l8.9 Parity II:Horizontal Velocities&Other Vector Components.·.······ 395 18.10The Pole Problem:Spherical Coordinates.................... 398 18.11Spherical Harmonics:Introduction........ 4 399 18.12Legendre Transforms and Other Sorrows 402 18.12.1 FFT in Longitude/MMT in Latitude .. 402 l8.12.2 Substitutes and Accelerators for the MMT...·.....···.··. 403 l8.l2.3 Parity and Legendre Transforms···.··········· 404vi CONTENTS 16 Coordinate Transformations 323 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 16.2 Programming Chebyshev Methods . . . . . . . . . . . . . . . . . . . . . . . . 323 16.3 Theory of 1-D Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 325 16.4 Infinite and Semi-Infinite Intervals . . . . . . . . . . . . . . . . . . . . . . . . 326 16.5 Maps for Endpoint & Corner Singularities . . . . . . . . . . . . . . . . . . . . 327 16.6 Two-Dimensional Maps & Corner Branch Points . . . . . . . . . . . . . . . . 329 16.7 Periodic Problems & the Arctan/Tan Map . . . . . . . . . . . . . . . . . . . . 330 16.8 Adaptive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 16.9 Almost-Equispaced Kosloff/Tal-Ezer Grid . . . . . . . . . . . . . . . . . . . . 334 17 Methods for Unbounded Intervals 338 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 17.2 Domain Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 17.2.1 Domain Truncation for Rapidly-decaying Functions . . . . . . . . . . 339 17.2.2 Domain Truncation for Slowly-Decaying Functions . . . . . . . . . . 340 17.2.3 Domain Truncation for Time-Dependent Wave Propagation: Sponge Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 17.3 Whittaker Cardinal or “Sinc” Functions . . . . . . . . . . . . . . . . . . . . . 341 17.4 Hermite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 17.5 Semi-Infinite Interval: Laguerre Functions . . . . . . . . . . . . . . . . . . . . 353 17.6 New Basis Sets via Change of Coordinate . . . . . . . . . . . . . . . . . . . . 355 17.7 Rational Chebyshev Functions: T Bn . . . . . . . . . . . . . . . . . . . . . . . 356 17.8 Behavioral versus Numerical Boundary Conditions . . . . . . . . . . . . . . 361 17.9 Strategy for Slowly Decaying Functions . . . . . . . . . . . . . . . . . . . . . 363 17.10Numerical Examples: Rational Chebyshev Functions . . . . . . . . . . . . . 366 17.11Semi-Infinite Interval: Rational Chebyshev T Ln . . . . . . . . . . . . . . . . 369 17.12Numerical Examples: Chebyshev for Semi-Infinite Interval . . . . . . . . . . 370 17.13Strategy: Oscillatory, Non-Decaying Functions . . . . . . . . . . . . . . . . . 372 17.14Weideman-Cloot Sinh Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 374 17.15Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 18 Spherical & Cylindrical Geometry 380 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 18.2 Polar, Cylindrical, Toroidal, Spherical . . . . . . . . . . . . . . . . . . . . . . 381 18.3 Apparent Singularity at the Pole . . . . . . . . . . . . . . . . . . . . . . . . . 382 18.4 Polar Coordinates: Parity Theorem . . . . . . . . . . . . . . . . . . . . . . . . 383 18.5 Radial Basis Sets and Radial Grids . . . . . . . . . . . . . . . . . . . . . . . . 385 18.5.1 One-Sided Jacobi Basis for the Radial Coordinate . . . . . . . . . . . 387 18.5.2 Boundary Value & Eigenvalue Problems on a Disk . . . . . . . . . . . 389 18.5.3 Unbounded Domains Including the Origin in Cylindrical Coordinates 390 18.6 Annular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 18.7 Spherical Coordinates: An Overview . . . . . . . . . . . . . . . . . . . . . . . 391 18.8 The Parity Factor for Scalars: Sphere versus Torus . . . . . . . . . . . . . . . 391 18.9 Parity II: Horizontal Velocities & Other Vector Components . . . . . . . . . . 395 18.10The Pole Problem: Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 398 18.11Spherical Harmonics: Introduction . . . . . . . . . . . . . . . . . . . . . . . . 399 18.12Legendre Transforms and Other Sorrows . . . . . . . . . . . . . . . . . . . . 402 18.12.1 FFT in Longitude/MMT in Latitude . . . . . . . . . . . . . . . . . . . 402 18.12.2 Substitutes and Accelerators for the MMT . . . . . . . . . . . . . . . . 403 18.12.3 Parity and Legendre Transforms . . . . . . . . . . . . . . . . . . . . . 404