Thomas Algorithm for Tridiagonal Systems

Transcription

1 Appendix Thomas Algorithm for Tridiagonal Systems A.I SALAR TRIDIAGONAL SYSTEMS For tridiagonal systems the LV decomposition method leads to an efficient algorithm, known as Thomas's algorithm. For a system of the form with akxk-l+bkxk+kxk+i=!k k=i,...,n (A.I) al = N = 0 (A.2) the following algorithm is obtained. Forward step {31=bl {3k=bk-ak- {3k-1 k-1 k=2,...,n 'YI-{31 -A 'Yk- _(-ak'yk-i+!k) {3k k - 2,..., N (A.3) Backward step XN = 'YN Xk='Yk-Xk+l~ k k=n-i,...,1 (A.4) This requires, in total, 5N operations. It can be shown that the above algorithm will always converge if the tridiagonal system is diagonal dominant, that is, if Ibkl~lakl+lckl Ibll>lcllandlbNI>laNI k=2,...,n-i (A.5) If a, b, are matrices we have a block-tridiagonal system, and the same algorithm can be applied. Due to the importance of triadiagonal system, we 505

2 506 present here a subroutine which can be used for an arbitrary scalar tridiagonal system. Subroutine TRIDAG SUBROUTINE TRIDAG (AA,BB,,FF,N1,N) ********************************************************************* SOLUTION OF A TRIDIAGONAL SYSTEM OF N-N1+1 EQUATIONS OF THE FORM AA(K)*X(K-1) + BB(K)*X(K) + (K)*X(K+1) = FF(K) K=N1,...,N K RANGING FROM N1 TO N THE SOLUTION X(K) IS STORED IN FF(K) AA (N1) AND (N) ARE NOT USED AA,BB,,FF ARE VETORS WITH DIMENSION N, TO BE SPEIFIED IN THE ALLING PROGRAM ********************************************************************* DIMENSION BB(N1)=1./BB(N1) AA(N1)=FF(N1)*BB(N1) N2=N1+1 N1N-N1+N DO 10 K-N2,N K1=K-1 (K1)=(K1)*BB(K1) BB(K) BB(K) AA(l),BB(l),(l),FF(l) =BB(K)-AA(K)*(K1) =l./bb(k) AA(K) =(FF(K)-AA(K)*AA(K1»*BB(K) 10 ONTINUE BAK SUBSTITUTION FF(N)-AA(N) DO 20 K1=N2,N K=N1N-K1 FF(K)-AA(K)-(K)*FF(K+1) 20 ONTINUE RETURN END A.2 PERIODI TRIDIAGONAL SYSTEMS For periodic boundary conditions, and a tridiagonal matrix with one in the extreme corners as in equation ( ), the above method does not apply. The following approach leads to an algorithm whereby two tridiagonal systems have to be solved. If the periodic matrix Bp( ii, E, c) has (N + 1) lines and columns resulting from a periodicity between points 1 and N + 2, the solution X is written as a linear combination X= X(I) + XN+IX(2), or Xk = xkl) + Xk2). XN+ I (A.6)

3 507 where xkl) and xk2) are solutions of the tridiagonal systems obtained by removing the last line and last column of Bp, containing the periodic elements. If this matrix is called B(N)(a, E, c) we solve successively, where the right-hand side terms Ik are put in a vector F: B(N)(a, E, )X(I) = F (A.7) and with B(N)(a, E, )X(2) = G (A.8) GT=(-OI,...,O, -N) (A.9) The last unknown XN+ I is obtained from the last equation by backsubstitution: I" (I) 0) _IN+I-N+IXr -ON+IXN (AIO) XN+ I - L. -..(2). -..(2). bn+i + ON+IXN + N+IXr The periodicity condition determines XN + 2 as Xn+2 = XI (A.II) The svbroutine TRIPER, based on this algorithm is included here. Note that if the periodicity condition is XN+2 = XI + (A.12) then the periodicity constant has to be added to the right-hand side of the last instruction, defining FF(N + 2). Subroutine TRIPER SUBROUTINE TRIPER(AA,BB,,FF,Nl,N,GAM2) ********************************************************************* SOLUTION OF A TRIDIAGONAL SYSTEM OF EQUATIONS WITH PERIODIITY IN THE POINTS K-Nl AND K~N+2 AA(K)*X(K-l) + BB(K)*X(K) + (K)*X(K+l) = FF(K) K=Nl,...,N+l THE ELEMENT IN THE UPPER RIGHT ORNER IS STORED IN AA(Nl) THE ELEMENT IN THE LOWER LEFT ORNER IS STORED IN (N+l) AA,BB,,FF,GAM2 ARE VETORS WITH DIMENSION N+2, TO BE SPEIFIED IN THE ALLING PROGRAM GAM2 IS AN AUXILIARY VETOR NEEDED FOR STORAGE THE SOLUTION IS STORED IN FF ********************************************************************* DIMENSION AA(1),BB(1),(1),FF(1),GAM2(1) BB(Nl)=l./BB(Nl) GAM2 (Nl)=-AA(Nl) AA(Nl)=FF(Nl)*BB(Nl) *BB(Nl)

4 508 N2=N1+1 N1N=N1+N DO 10 K=N2,N K1-K-1 (K1)=(K1)*BB(K1) BB(K) =BB(K)-AA(K)*(K1) BB(K) =l./bb(k) GAM2 (K)=-AA(K) *GAM2 (K1)*BB(K) AA(K) =(FF(K)-AA(K)*AA(K1»*BB(K) 10 ONTINUE GAM2 (N)=GAM2(N)-(N) *BB(N) BAK SUBSTITUTION FF(N)=AA(N) BB(N)=GAM2(N) DO 20 K1=N2,N K=N1N-K1.'- K2=K+1 FF(K)-AA(K)-(K)*FF(K2) BB(K)=GAM2 (K)-(K) *BB(K2) 20 ONTINUE K1=N+1 ZAA=FF(K1)-(K1) *FF(N1)-AA(K1) *FF(N) ZAA=ZAA/(BB(K1)+AA(K1) *BB(N)+(K1) *BB(N1» FF(K1)=ZAA DO 30 K=N1,N FF(K)=FF(K)+BB(K)*ZAA 30 ONTINUE FF(N+2)=FF(N1) RETURN END

5 Index Accuracy 161, 166, 265, 274, 275 Brailowskaya method 433 order of accuracy 161, 164, 198,219, Briley and McDonald method ,276,278,342,346,354, Burger's equation 269, 326, 327, 360, 356, ,364,365,436,437 ADl method 424, 437, 439, 440, 442, 457, 481 auchy problem 153, 409 Algebriac system 161, 163,421,456, ebeci-smith model 50, 51, 56c~\\-c~..\-nd fw~."..i\i 500 entrifugal force 17 14"f, 24- Aliasing phenomenon 325 haracteristic direction Amplification factor of numerical normal scheme 287, 296, 336, 375, 379, polynomial 377, 391, 393, 394, 425, 401,402,409, ,428,432,446 of differential equation 301, 343, 346, speed ,375 surface Amplification matrix 296, 336 irculation of iterative method 463, 464, 466, ircular cylinder, flow over ,470,471,478 lebsch representation Amplitude error 303, 353, 380, 444 ollocation 203, 223 Approximate factorization (see ompact differencing formula Factorization) ompatibility relations (equations) Approximation level (see Level of approximation) ondition number 373, 479 Artificial dissipation (viscosity) 324, 326, onditional stability 274, , 397, 398, 399 onditioning operator (see onvergence A-stability 423, 427, 429 operator) Averaging procedures for turbulent flow onforming element 205 (see Turbulent flow) onjugate gradient method 457, 484 onservation form 13, 19,88,108,139, Banded matrix , 238 Beam and Warming scheme 359, 360, onservation law 8, 12, 221, 223, 237, 426,430,431,436, Boundary conditions 153, 156, 267, 268, differential form ,409,410,412,413,442 for mass 12 Dirichlet 155,406,442,447,448,458, for momentum 14,15,16, for energy 18,24,111 Neumann 155, 381, 383, 385, 457 general form 10, 13 periodic 370, 376, 382, 385, 388, 393, integral form scalar 9 Boundary layer approximation 63, 64, vector 11 75, 78, 79 onservative discretization (scheme) Boundary-layer equations 154,

6 510 onservative discretization (scheme) Discretization error 248, 265, , (cont.) 301 variables 29,88 operator 290, 296 onsistency (condition) 267,275,276, Dispersion error 303,317,344,346, 277,278,345,348,350,351,356, 353,354,360,364,380, , 362, 363, 376, 379, 425, 445 Dispersion relation ontact discontinuity 89, 90 exact 152, 302 ontinuity equation 108 numerical 302, 318 ontrol-volume 237, 241, 253 Displacement operator 171 onvection-diffusion equation 268, Dissipation (viscous) , 270, 322, 329, 331, 334, 336, Dissipation error (see Diffusion error) 403, 409, 411 numerical (see Numerical dissipation) onvection equation 267, 270, 272, 289, Dissipation in the sense of Kreiss 324, 303,305,319,321,342,347,351, 352,375,379,380,382,386,392, 325,354,360 Dissipative scheme 324, 325, ,394,395,400,405,411,412, Distributed loss model , 429, 430, 433, 435, 442, 443, friction force 82, , 481 Divergence form 13 onvergence (condition) 162, 166, 275, Domain of dependence 143, Douglas and Rachford method 439 of iterative method 457, Du Fort and Frankel scheme 313,314, matrix (operator) 462, 478, 485, rate , 468, 480 oriolis force 16 ourant-friedrichs-lewy (FL) Elliptic equation 133, ,269, condition 287, 288, 289, 305, 353, 300,301,421, ,394,405,433,434,448 Energy (conservation) equation 19, III, ourant number 272, 288, 308, 314, ,319,326,327,328,360,364, Enthalpy 19, , 448 stagnation (total) enthalpy 19, 32, ran4k-nic 0Ison method 426, 429, 109, ,440 Entropy 20,21,31-33,108,110, III rocco's equation 21, 24, 82, 88 condition 92, 120 urved duct 72 equation 20, 83, 85, 88 inequality 92 Damping 305,317,323,346,389,444, production ,467,468,470,480,488,489, Equation discretization 161, f Equation of state 32 Delta form 436, 439 Equivalence Theorem of Lax 281, 401 Diagonal dominance 194, 472 Equivalent differential equation 265, Diffusion error 303, 316, 317, 344, 346, 277,278, , , 430 Error analysis for hyperbolic problems Diffusion equation 268,291,303,314, , ,330,343,380,382,385,392, for parabolic problems , 395, 405, 429, 437, 440,,448 Error of iterative scheme 462, 463, 465, Diffusion phenomena 133, 135, 268, , 406 Error of numerical scheme 283, 284, Direct method 163,456,463, ,412,413,415 Dirichlet boundary condition (see high-frequency 304, 310, 313, 316, Boundary condition) 319, 325, 326, 354, 430, 449, Discontinuities 13, 88, 92, 111,311, 467,468, ,316,317,365 low-frequency 304,316,318,325, Discrete Fourier transform , 467, 468,

7 Euler equations 15, 87-99, 111, 125, Gauss and Seidel point iteration 460, 128,240,298,386,397,423,429, 437, 447, 448, ,462,464,466,468,469,471, 473, 486 Euler explicit method 271, 289, 298, line iteration ,388, ,395,403,413, Gaussian quadrature , 429, 433, 478, 481 Generating polynomial 425, 426 Euler implicit method 271, 289, 379, Godunov and Ryabenkii condition ,394,429,439 Group velocity 152, , 327, 409 EVP method 301 (', Expansion shock 91 Heat conduction equation 268 i Explicit method (scheme) 162,271, 376, Heat sources , 431, 432 Helmholtz equation 16, 44 Henn's method 433, 446 \\ Factorization 424, 439, 440 Hermitian elements 205, Fick's law 133 schemes 181, 183 Finite difference method , 319, Horizontal line relaxation , 383 Hybrid equations 140, 154 Finite difference operators 168, 171, Hyperbolic equation (problem) 133, 172,174, averaging operator ,267,270,303,305,322, 323, ,386,387,409,421, backward difference 168, 172, 448, ,271,389,394 non-linear central difference 168,172, , 331, 336, 343, 374, 378, 388, III-posed problem ,392,395,403,423,429,433 Implicit difference formulas 171, forward difference 168,172, , ,272, 389, 391 Implicit method 162, 163, 376, 423, for mixed derivatives ,437 on non-uniform mesh Implicit scheme 271, 279, 280, 412 Finite element method 125, 190,201, Incomplete holeski factorization ,382,459,484 Incompressible fluid 44,45,106,127 Finite volume method 223, 224, 237, Initial (boundary) value problem ,283,301,370,371.,386 Five-point formula for Laplace equation Integral formulation ,301,457 Internal energy 20 Flux 9, 10, 29, , 360, 436 Internal flows convective 9, 11, 18 Interior scheme 387, 403, 409 diffusive 9, 11, 18 Interpolation functions 203, Fourier analysis (decomposition) Inviscid flow 84, 100, ,290,320,375,385 Inviscid pressure field harmonics 285,290,315,323,344, Irregular mesh Irrotational flows 108, 109, 120 of iterative scheme Isentropic condition Fourier's law 18 potential equation Fractional-step method 438, 439, 444 shock Friction force 83, 84 Isoparametric transformaticjp (mapping) Fromm's scheme , 227 Iterative method (scheme) 163;412, Galerkin method 190, 218, , \ Galerkin and Bubnow method 218 Galerkin and Petrov method 218, 222 Jacobi point iterative method 412, Gas constant , 464, 466, 467, 469, 472, Gauss points

8 512 Jacobi point iterative method (cont.) Multistep method , line relaxation 475, 476, 487, 488 Jacobian matrix 139, 150, 436, 485 Navier-Stokes equations 15,29-48,63, of flux vector 139, ,154,424,428,433,435,488 of isoparametric transformation Neumann boundary condition (see Boundary condition) Kutta-Joukowski condition , Newton method 435, Newtonian fluid 14, 15, 16,31, Nine-point formula for Laplace Lagrangian elements , equation 190, 191 Non-conservative form (scheme) 15, Laminar flow 45, 71, , 239 Laplace's equation ,219,301, Non-dimensional form of equations 437,471,472,479,486,487,490 Lax equivalence theorem ,134 ( Non-linear error terms 343, Lax-Friedrichs scheme 299,306,315, instability 326,327,328,397, ,324,333,348,349,352,354, Non-uniform mesh , 244, Non-uniqueness of potential flow Lax-Wendroff scheme ,314, ,317,325,348,354,355,357, of viscous flow , 364, 435 Normal matrix 297, 321, 401 Leapfrog scheme , , Normal mode representation 266, 326,336,347,348,357,366,374, ,388,390,393,395,413,423 Levels of approximation 133, 135, 161 Nozzle Numerical boundary condition 265, 319 dynamical level 5, 26 Numerical dissipation 307, 324, 326, spatial level 4 329, 346, 347, 358, 360, 364, 365, steadiness level 4 366,379,389,390,397,431,443, Linearization 431, Linearized potential flow 127 Numerical domain of dependence 289 Local mode analysis 375 Numerical flux 241, 361, 364 Mach angle 148 Numerical viscosity 277, 346, 352, 353, 358, 363, 364 Mach line 137 Mach number 32,35,44,51,56,67, Odd-even decoupling (oscillations) 189 6?, 81, 95, 97, ,137 Order of accuracy 161,167,197,219, Mappmg 2~6 Mass lump!ng , , 343, 346, 351, 356, 360 Mass. matnx Ordinary differential equations Matnx form (of dlscretlzed equations) Oscillations 311, 406, , 290 non-linear , 365 method 265, Overrelaxation Mcormack's met~od 433, 434, 437 Gauss-Seidel Mesh Reynolds (Peclet) number 334, Jacobi ~, 410, 411 successive point (SOR) 471,473,487 Method of lines 271 successive line 474 Mixed derivatives symmetric Modal equation 374, 425, 432 Momentum conservation Momentum equation 108, 112 Panel method 128 Monotone scheme 357 Parabolic equation (problem) 133, 154, Multigrid method 163, 422, 457, 467, 267, 268, 270, 303, 304, 320, 322, ,421,437

9 513 Parabolized Navier-Stokes equations Rothalpy 21, 24, 85, 100 (PNS) 70-75, 421 Round-off error 284, 411 Peaceman-Rachford method 441 Runge-Kutta method 413, 424, Peclet number 268 Perfect gas 31-33, 103 Secondary flows 74 Periodic boundary conditions (see Shallow-water equations 141, 150,291, Boundary condition) 298 Phase angle 286,316 Shear stress 14,18 Phase error (see Dispersion error) Shock 35, 44, 51, 69189, 90, 91, lagging 305, 308, 316, 317, 360, ,240,318,326 leading 305, 307, 353, 360 normal shock Phase velocity 149, 153, 303, 318, 319 Singularity method 127 Physical boundary condition Slip lines 91 Poisson equation , 466, 488 Small disturbance approximation 126, for pressure ,147 Potential flow , 137, 155, 269, Sonic velocity 33, , 482 Space discretization 161 Potential shocks eigenvalues of 371,373,377,488 Prandtl number 19 operator 290, , 425, 436, 456, Preconditioning 163, 374, 422, 457, spectrum of , 488 Predictor-corrector method , stability of 373, Space marching 412 Pressure correction method 73 Spacelike variables Principal root 379 Specific heat, ratio of 18, 32, 103 Propagatidh phenomena 134, 386 Spectral analysis of numerical errors Pseudo-unsteady formulation 162, ,422 radius 296, 297, 321, 400, 402, 406, Quasi-linear differential equations 133, 408, 413, 463, 468, 469, 472, Spectrum (of a difference operator) Quasi-three-dimensional approximation (see Levels of approximation) for convection equation for diffusion equation 378, 380, 385, Rankine-Hugoniot relations 89-93, , 123, 240 of iterative method Relaxation method 457, 488 Speed of sound (see Sonic velocity) equivalent differential equation Spline methods Spurious solution (root) 314, 379 parameter 471,479,482 red-black 476, 492 Stability (condition) 162, 166,265,275, 277, ,287,342,343,345, smoothing properties ,356,357,375,376,377, zebra 477,478, ,400,402,403,423,425, Residual 216, 278, 461, 462, ,429,432,446 Reynolds-averaged Navier-Stokes for convection-diffusion equation equations 48, , Reynolds number 45,75,87, 134,268, for hyperbolic problems , Reynolds stress 50 of iterative method 478 Richardson's method 479, 482 from matrix method , Rotary stagnation pressure 106, Rotating frame of reference 16-18, 82, for multi-dimensional problems Rotational flow 24, neutral 313,326

10 514 Stability (condition) (cont.) Transonic flow , 137, 240, 270, for non-linear problems for parabolic problems 303, 320, Transport equation 268, Trapezoidal method 426, 428, 430 from Von Neumann method , Tridiagonal system 182, 183, 271, 346,351,352,355,356,359, 370,375,385,392, ,430,431,437,441,460 eigenvalues of Stagnation enthalpy (see Enthalpy) Truncation error 174, 175, , Stagnation pressure 33,103, ,240,265, ,342,343, Stagnation temperature ,351,352,361,362,390,464 Steady-state (stationary) formulation 70, Turbulence models 51 84, 373, , 421, 424, 444, Turbulent flow 47, 48, 50-62, 73, , 479 Two-level schemes (general form) inviscid flows , , 375,400, potential flow ,431 Stiffness matrix 218, 484 Stokes Streamline equation 108, Unconditional instability 287 ' 393, 405, Strongl~ implicit procedure (SIP) 483 stabili~y 289, 314, 345, 413, 423, 428, Subsonic flow 120, 137, 155, 270, Successive overrelaxation (SOR) (see Unstable s~heme 272 Supercrit~cal Overrelaxation) airfoil 119 Upwind 351,354,358,386,389,392,395, scheme ' Supersonic flow 137, 270, Sutherland's formula 31 Variational principle (formulation) 204, Taylor expansion , 174, 183, ' 361' 379' 425' 426' 432', Vertical line relaxation 476, 488 Viscid-inviscid interaction 78 Test fu~ctio~s 217 " Viscosity, coefficient of 14, 15,31 Thermal conductivity, coefficient of 19, Viscous-inviscid interaction 27,87 31 Von Neumann method 265, , Thermal diffusivity ,351,355,356,359,370,375, Thin-shear layer approximation 27, 385, 392, 393, 401, 402, , Thomas al~ori~hm 457, ,413,422,434,441,468 Von Neumann polynomial Three-level schemes ' 378, 379, Vortex Vortex sheet singularity 89, Through-flo~ 84 ' Vorticity equation 15, 44, 93 Time-dependent (equation) approach vector 15, 18, 24, 88, 100, 102, ,270,373,412,421,423,456, 464. WarmIng and Beam scheme (see Beam Time-integration method 370, 374, 413, and Warming scheme) 456 stability of , , 413, Wave equation 270, 291, 300, 314, Wave-front 135,136,140,146 Time-like variable ,421 Wave-length 285, 286 Time-marching method 162 Wave-like solutions 135, 136, , Total energy Total enthalpy (see Enthalpy) Wave-number , 285, Transient solution 373,374,376,377, vector , 466 Wave packet , 360

11 Wave (phase) speed 149, 299, 305 Well-posed problem (in the sense of Weak formulation 204, 217, 221 Hadamard) 152, 343, 372, 377, Weak instability , 394, 486 Weak solution 241 Weighted residual method 204, Zone of dependence J,:;:zi~1 ~ ",.-, "",.. r j,'*; ;~' -"

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