| 1 |
C ------------
|
| 2 |
CN NAME: R I E M A N N _ V T
|
| 3 |
C ------------
|
| 4 |
|
| 5 |
CP PURPOSE:
|
| 6 |
CP THIS PROGRAM COMPUTES THE SOLUTION OF A 1D
|
| 7 |
CP RELATIVISTIC RIEMANN PROBLEM WITH ARBITRARY TANGENTIAL VELOCITIES
|
| 8 |
CP (FOR CONSTANT-GAMMA IDEAL GASES)
|
| 9 |
CP WITH INITIAL DATA UL IF X<R0 AND UR IF X>R0
|
| 10 |
CP IN THE WHOLE SPATIAL DOMAIN [R0 - 0.5,R0 + 0.5]
|
| 11 |
C
|
| 12 |
|
| 13 |
CC COMMENTS:
|
| 14 |
CC SEE PONS, MARTI AND MUELLER, JFM, 2000
|
| 15 |
CC
|
| 16 |
CC WRITTEN BY: Jose-Maria Marti
|
| 17 |
CC Departamento de Astronomia y Astrofisica
|
| 18 |
CC Universidad de Valencia
|
| 19 |
CC 46100 Burjassot (Valencia), Spain
|
| 20 |
CC jose-maria.marti@uv.es
|
| 21 |
CC AND
|
| 22 |
CC Ewald Mueller
|
| 23 |
CC Max-Planck-Institut fuer Astrophysik
|
| 24 |
CC Karl-Schwarzschild-Str. 1
|
| 25 |
CC 85741 Garching, Germany
|
| 26 |
CC emueller@mpa-garching.mpg.de
|
| 27 |
C
|
| 28 |
|
| 29 |
PROGRAM RIEMANN_VT
|
| 30 |
|
| 31 |
IMPLICIT NONE
|
| 32 |
|
| 33 |
INCLUDE 'npoints'
|
| 34 |
|
| 35 |
C ------
|
| 36 |
C COMMON BLOCKS
|
| 37 |
C ------
|
| 38 |
|
| 39 |
DOUBLE PRECISION RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 40 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 41 |
COMMON /STATES/ RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 42 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 43 |
|
| 44 |
DOUBLE PRECISION RHOLS, ULS, HLS, CSLS, VELLS, VELTLS, VSHOCKL
|
| 45 |
COMMON /LS/ RHOLS, ULS, HLS, CSLS, VELLS, VELTLS, VSHOCKL
|
| 46 |
|
| 47 |
DOUBLE PRECISION RHORS, URS, HRS, CSRS, VELRS, VELTRS, VSHOCKR
|
| 48 |
COMMON /RS/ RHORS, URS, HRS, CSRS, VELRS, VELTRS, VSHOCKR
|
| 49 |
|
| 50 |
DOUBLE PRECISION GB
|
| 51 |
COMMON /GB/ GB
|
| 52 |
|
| 53 |
DOUBLE PRECISION QX(NG), QW(NG)
|
| 54 |
COMMON /INT/ QX, QW
|
| 55 |
|
| 56 |
C -----------
|
| 57 |
C INTERNAL VARIABLES
|
| 58 |
C -----------
|
| 59 |
|
| 60 |
INTEGER MN, N, I, ILOOP
|
| 61 |
PARAMETER (MN = 400)
|
| 62 |
|
| 63 |
DOUBLE PRECISION TOL, PMIN, PMAX, DVEL1, DVEL2, CHECK, R0
|
| 64 |
PARAMETER (R0=0.D0)
|
| 65 |
|
| 66 |
DOUBLE PRECISION PS, VELS
|
| 67 |
|
| 68 |
DOUBLE PRECISION RHOA(MN), PA(MN), VELA(MN), VELTA(MN),
|
| 69 |
& UA(MN), HA(MN)
|
| 70 |
|
| 71 |
DOUBLE PRECISION A
|
| 72 |
|
| 73 |
DOUBLE PRECISION RAD(MN), X1, X2, X3, X4, X5, T, LAM, LAMS
|
| 74 |
|
| 75 |
DOUBLE PRECISION KL, KR, CONST
|
| 76 |
|
| 77 |
C ----------
|
| 78 |
C INITIAL STATES
|
| 79 |
C ----------
|
| 80 |
|
| 81 |
WRITE(*,*) ' ADIABATIC INDEX OF THE GAS: '
|
| 82 |
READ (*,*) GB
|
| 83 |
|
| 84 |
WRITE(*,*) ' TIME FOR THE SOLUTION: '
|
| 85 |
READ (*,*) T
|
| 86 |
|
| 87 |
C -----
|
| 88 |
C LEFT STATE
|
| 89 |
C -----
|
| 90 |
|
| 91 |
WRITE(*,*) ' ---- LEFT STATE ---- '
|
| 92 |
WRITE(*,*) ' PRESSURE : '
|
| 93 |
READ (*,*) PL
|
| 94 |
WRITE(*,*) ' DENSITY : '
|
| 95 |
READ (*,*) RHOL
|
| 96 |
WRITE(*,*) ' FLOW VELOCITY-X: '
|
| 97 |
READ (*,*) VELL
|
| 98 |
WRITE(*,*) ' FLOW VELOCITY-Y: '
|
| 99 |
READ (*,*) VELTL
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| 100 |
|
| 101 |
C ------
|
| 102 |
C RIGHT STATE
|
| 103 |
C ------
|
| 104 |
|
| 105 |
WRITE(*,*) ' ---- RIGHT STATE --- '
|
| 106 |
WRITE(*,*) ' PRESSURE : '
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| 107 |
READ (*,*) PR
|
| 108 |
WRITE(*,*) ' DENSITY : '
|
| 109 |
READ (*,*) RHOR
|
| 110 |
WRITE(*,*) ' FLOW VELOCITY-X: '
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| 111 |
READ (*,*) VELR
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| 112 |
WRITE(*,*) ' FLOW VELOCITY-Y: '
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| 113 |
READ (*,*) VELTR
|
| 114 |
|
| 115 |
|
| 116 |
C ---------------------------------------
|
| 117 |
C SPECIFIC INTERNAL ENERGY, SPECIFIC ENTHALPY, SOUND SPEED,
|
| 118 |
C ADIABATIC CONSTANT AND FLOW LORENTZ FACTORS IN THE INITIAL STATES
|
| 119 |
C ---------------------------------------
|
| 120 |
|
| 121 |
UL = PL/(GB - 1.D0)/RHOL
|
| 122 |
UR = PR/(GB - 1.D0)/RHOR
|
| 123 |
|
| 124 |
HL = 1.D0 + GB*UL
|
| 125 |
HR = 1.D0 + GB*UR
|
| 126 |
|
| 127 |
CSL= DSQRT((GB - 1.D0)*(HL - 1.D0)/HL)
|
| 128 |
CSR= DSQRT((GB - 1.D0)*(HR - 1.D0)/HR)
|
| 129 |
|
| 130 |
KL = PL/RHOL**GB
|
| 131 |
KR = PR/RHOR**GB
|
| 132 |
|
| 133 |
WL = 1.D0/DSQRT(1.D0 - VELL*VELL - VELTL*VELTL)
|
| 134 |
WR = 1.D0/DSQRT(1.D0 - VELR*VELR - VELTR*VELTR)
|
| 135 |
|
| 136 |
C ------
|
| 137 |
C COEFFICIENTS FOR NUMERICAL INTEGRATION IN RAREFACTIONS
|
| 138 |
C ------
|
| 139 |
|
| 140 |
CALL GAULEG(-1.D0,1.D0,QX,QW,NG)
|
| 141 |
|
| 142 |
C --------
|
| 143 |
C NUMBER OF POINTS
|
| 144 |
C --------
|
| 145 |
|
| 146 |
N = 400
|
| 147 |
|
| 148 |
C -------------
|
| 149 |
C TOLERANCE FOR THE SOLUTION
|
| 150 |
C -------------
|
| 151 |
|
| 152 |
TOL = 0.D0
|
| 153 |
|
| 154 |
C
|
| 155 |
|
| 156 |
ILOOP = 0
|
| 157 |
|
| 158 |
IF ((PL.EQ.PR).AND.(VELL.EQ.VELR)) THEN
|
| 159 |
|
| 160 |
PS = PL
|
| 161 |
VELS = VELL
|
| 162 |
|
| 163 |
VSHOCKL = VELL
|
| 164 |
RHOLS = (PS/KL)**(1.D0/GB)
|
| 165 |
ULS = PS/(GB - 1.D0)/RHOLS
|
| 166 |
|
| 167 |
VSHOCKR = VELL
|
| 168 |
RHORS = (PS/KR)**(1.D0/GB)
|
| 169 |
URS = PS/(GB - 1.D0)/RHORS
|
| 170 |
|
| 171 |
ELSE
|
| 172 |
|
| 173 |
PMIN = (PL + PR)/2.D0
|
| 174 |
PMAX = PMIN
|
| 175 |
|
| 176 |
5 ILOOP = ILOOP + 1
|
| 177 |
|
| 178 |
PMIN = 0.5D0*MAX(PMIN,0.D0)
|
| 179 |
PMAX = 2.D0*PMAX
|
| 180 |
|
| 181 |
|
| 182 |
CALL GETDVEL2(PMIN, DVEL1)
|
| 183 |
|
| 184 |
CALL GETDVEL2(PMAX, DVEL2)
|
| 185 |
|
| 186 |
CHECK = DVEL1*DVEL2
|
| 187 |
IF (CHECK.GT.0.D0) GOTO 5
|
| 188 |
|
| 189 |
|
| 190 |
C ---------------------------
|
| 191 |
C PRESSURE AND FLOW VELOCITY IN THE INTERMEDIATE STATES
|
| 192 |
C ---------------------------
|
| 193 |
|
| 194 |
CALL GETP2(PMIN, PMAX, TOL, PS)
|
| 195 |
|
| 196 |
VELS = 0.5D0*(VELLS + VELRS)
|
| 197 |
|
| 198 |
WRITE(*,*) 'VELS = ', VELS, 'PS = ', PS
|
| 199 |
|
| 200 |
ENDIF
|
| 201 |
|
| 202 |
C -------
|
| 203 |
C SOLUTION ON THE NUMERICAL MESH
|
| 204 |
C -------
|
| 205 |
|
| 206 |
C -----------
|
| 207 |
C POSITIONS OF THE WAVES
|
| 208 |
C -----------
|
| 209 |
|
| 210 |
IF (PL.GE.PS) THEN
|
| 211 |
|
| 212 |
CONST = HL*WL*VELTL
|
| 213 |
|
| 214 |
CALL FLAMB(KL, CONST, PL, VELL, 'L', LAM )
|
| 215 |
|
| 216 |
CALL FLAMB(KL, CONST, PS, VELS, 'L', LAMS)
|
| 217 |
|
| 218 |
X1 = R0 + LAM *T
|
| 219 |
X2 = R0 + LAMS*T
|
| 220 |
|
| 221 |
ELSE
|
| 222 |
|
| 223 |
X1 = R0 + VSHOCKL*T
|
| 224 |
X2 = X1
|
| 225 |
|
| 226 |
END IF
|
| 227 |
|
| 228 |
X3 = R0 + VELS*T
|
| 229 |
|
| 230 |
IF (PR.GE.PS) THEN
|
| 231 |
|
| 232 |
CONST = HR*WR*VELTR
|
| 233 |
|
| 234 |
CALL FLAMB(KR, CONST, PS, VELS, 'R', LAMS)
|
| 235 |
|
| 236 |
CALL FLAMB(KR, CONST, PR, VELR, 'R', LAM )
|
| 237 |
|
| 238 |
X4 = R0 + LAMS*T
|
| 239 |
X5 = R0 + LAM *T
|
| 240 |
|
| 241 |
ELSE
|
| 242 |
|
| 243 |
X4 = R0 + VSHOCKR*T
|
| 244 |
X5 = X4
|
| 245 |
|
| 246 |
END IF
|
| 247 |
|
| 248 |
C ----------
|
| 249 |
C SOLUTION ON THE MESH
|
| 250 |
C ----------
|
| 251 |
|
| 252 |
DO 10 I=1,N
|
| 253 |
|
| 254 |
RAD(I) = R0 + DFLOAT(I)/DFLOAT(N) - 0.5D0
|
| 255 |
|
| 256 |
10 CONTINUE
|
| 257 |
|
| 258 |
DO 120 I=1,N
|
| 259 |
|
| 260 |
IF (RAD(I).LE.X1) THEN
|
| 261 |
|
| 262 |
PA(I) = PL
|
| 263 |
RHOA(I) = RHOL
|
| 264 |
VELA(I) = VELL
|
| 265 |
VELTA(I) = VELTL
|
| 266 |
UA(I) = UL
|
| 267 |
HA(I) = 1.D0 + GB*UA(I)
|
| 268 |
|
| 269 |
ELSE IF (RAD(I).LE.X2) THEN
|
| 270 |
|
| 271 |
A = (RAD(I) - R0)/T
|
| 272 |
|
| 273 |
CALL RAREF2(A, PS, RHOL, PL, UL, CSL, VELL, VELTL,
|
| 274 |
& 'L', RHOA(I), PA(I), UA(I), VELA(I), VELTA(I))
|
| 275 |
|
| 276 |
ELSE IF (RAD(I).LE.X3) THEN
|
| 277 |
|
| 278 |
PA(I) = PS
|
| 279 |
RHOA(I) = RHOLS
|
| 280 |
VELA(I) = VELS
|
| 281 |
UA(I) = ULS
|
| 282 |
HA(I) = 1.D0 + GB*UA(I)
|
| 283 |
CONST = HL*WL*VELTL
|
| 284 |
VELTA(I) = CONST*DSQRT((1.D0 - VELS*VELS)/
|
| 285 |
& (CONST**2 + HA(I)**2))
|
| 286 |
|
| 287 |
ELSE IF (RAD(I).LE.X4) THEN
|
| 288 |
|
| 289 |
PA(I) = PS
|
| 290 |
RHOA(I) = RHORS
|
| 291 |
VELA(I) = VELS
|
| 292 |
UA(I) = URS
|
| 293 |
HA(I) = 1.D0 + GB*UA(I)
|
| 294 |
CONST = HR*WR*VELTR
|
| 295 |
VELTA(I) = CONST*DSQRT((1.D0 - VELS*VELS)/
|
| 296 |
& (CONST**2 + HA(I)**2))
|
| 297 |
|
| 298 |
ELSE IF (RAD(I).LE.X5) THEN
|
| 299 |
|
| 300 |
A = (RAD(I) - R0)/T
|
| 301 |
|
| 302 |
CALL RAREF2(A, PS, RHOR, PR, UR, CSR, VELR, VELTR,
|
| 303 |
& 'R', RHOA(I), PA(I), UA(I), VELA(I), VELTA(I))
|
| 304 |
|
| 305 |
ELSE
|
| 306 |
|
| 307 |
PA(I) = PR
|
| 308 |
RHOA(I) = RHOR
|
| 309 |
VELA(I) = VELR
|
| 310 |
VELTA(I) = VELTR
|
| 311 |
UA(I) = UR
|
| 312 |
HA(I) = 1.D0 + GB*UA(I)
|
| 313 |
|
| 314 |
END IF
|
| 315 |
|
| 316 |
120 CONTINUE
|
| 317 |
|
| 318 |
OPEN (3,FILE='solution.dat',FORM='FORMATTED',STATUS='NEW')
|
| 319 |
|
| 320 |
WRITE(3,150) N, T
|
| 321 |
150 FORMAT(I5,1X,F10.5)
|
| 322 |
|
| 323 |
DO 60 I=1,N
|
| 324 |
WRITE(3,200) RAD(I),PA(I),RHOA(I),VELA(I),VELTA(I),UA(I)
|
| 325 |
60 CONTINUE
|
| 326 |
|
| 327 |
200 FORMAT(5(E14.8,1X))
|
| 328 |
|
| 329 |
CLOSE(3)
|
| 330 |
|
| 331 |
STOP
|
| 332 |
END
|
| 333 |
|
| 334 |
C ----------
|
| 335 |
CN NAME: G E T D V E L 2
|
| 336 |
C ----------
|
| 337 |
|
| 338 |
CP PURPOSE:
|
| 339 |
CP COMPUTE THE DIFFERENCE IN FLOW SPEED BETWEEN LEFT AND RIGHT INTERMEDIATE
|
| 340 |
CP STATES FOR GIVEN LEFT AND RIGHT STATES AND PRESSURE
|
| 341 |
C
|
| 342 |
|
| 343 |
CC COMMENTS:
|
| 344 |
CC NONE
|
| 345 |
|
| 346 |
SUBROUTINE GETDVEL2( P, DVEL )
|
| 347 |
|
| 348 |
IMPLICIT NONE
|
| 349 |
|
| 350 |
INCLUDE 'npoints'
|
| 351 |
|
| 352 |
C ------
|
| 353 |
C ARGUMENTS
|
| 354 |
C ------
|
| 355 |
|
| 356 |
DOUBLE PRECISION P, DVEL
|
| 357 |
|
| 358 |
C ------
|
| 359 |
C COMMON BLOCKS
|
| 360 |
C ------
|
| 361 |
|
| 362 |
DOUBLE PRECISION RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 363 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 364 |
COMMON /STATES/ RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 365 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 366 |
|
| 367 |
DOUBLE PRECISION RHOLS, ULS, HLS, CSLS, VELLS, VELTLS, VSHOCKL
|
| 368 |
COMMON /LS/ RHOLS, ULS, HLS, CSLS, VELLS, VELTLS, VSHOCKL
|
| 369 |
|
| 370 |
DOUBLE PRECISION RHORS, URS, HRS, CSRS, VELRS, VELTRS, VSHOCKR
|
| 371 |
COMMON /RS/ RHORS, URS, HRS, CSRS, VELRS, VELTRS, VSHOCKR
|
| 372 |
|
| 373 |
DOUBLE PRECISION GB
|
| 374 |
COMMON /GB/ GB
|
| 375 |
|
| 376 |
DOUBLE PRECISION QX(NG), QW(NG)
|
| 377 |
COMMON /INT/ QX, QW
|
| 378 |
|
| 379 |
C -----
|
| 380 |
C LEFT WAVE
|
| 381 |
C -----
|
| 382 |
|
| 383 |
CALL GETVEL2(P, RHOL, PL, UL, HL, CSL, VELL, VELTL, WL, 'L',
|
| 384 |
& RHOLS, ULS, HLS, CSLS, VELLS, VELTLS, VSHOCKL)
|
| 385 |
|
| 386 |
C -----
|
| 387 |
C RIGHT WAVE
|
| 388 |
C -----
|
| 389 |
|
| 390 |
CALL GETVEL2(P, RHOR, PR, UR, HR, CSR, VELR, VELTR, WR, 'R',
|
| 391 |
& RHORS, URS, HRS, CSRS, VELRS, VELTRS, VSHOCKR)
|
| 392 |
|
| 393 |
DVEL = VELLS - VELRS
|
| 394 |
|
| 395 |
RETURN
|
| 396 |
END
|
| 397 |
C -------
|
| 398 |
CN NAME: G E T P 2
|
| 399 |
C -------
|
| 400 |
|
| 401 |
CP PURPOSE:
|
| 402 |
CP FIND THE PRESSURE IN THE INTERMEDIATE STATE OF A RIEMANN PROBLEM IN
|
| 403 |
CP RELATIVISTIC HYDRODYNAMICS
|
| 404 |
C
|
| 405 |
|
| 406 |
CC COMMENTS:
|
| 407 |
CC THIS ROUTINE USES A COMBINATION OF INTERVAL BISECTION AND INVERSE
|
| 408 |
CC QUADRATIC INTERPOLATION TO FIND THE ROOT IN A SPECIFIED INTERVAL.
|
| 409 |
CC IT IS ASSUMED THAT DVEL(PMIN) AND DVEL(PMAX) HAVE OPPOSITE SIGNS WITHOUT
|
| 410 |
CC A CHECK.
|
| 411 |
CC ADAPTED FROM "COMPUTER METHODS FOR MATHEMATICAL COMPUTATION",
|
| 412 |
CC BY G. E. FORSYTHE, M. A. MALCOLM, AND C. B. MOLER,
|
| 413 |
CC PRENTICE-HALL, ENGLEWOOD CLIFFS N.J.
|
| 414 |
|
| 415 |
SUBROUTINE GETP2( PMIN, PMAX, TOL, PS )
|
| 416 |
|
| 417 |
IMPLICIT NONE
|
| 418 |
|
| 419 |
C -----
|
| 420 |
C ARGUMENTS
|
| 421 |
C -----
|
| 422 |
|
| 423 |
DOUBLEPRECISION PMIN, PMAX, TOL, PS
|
| 424 |
|
| 425 |
C -------
|
| 426 |
C COMMON BLOCKS
|
| 427 |
C -------
|
| 428 |
|
| 429 |
DOUBLE PRECISION GB
|
| 430 |
COMMON /GB/ GB
|
| 431 |
|
| 432 |
DOUBLE PRECISION RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 433 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 434 |
COMMON /STATES/ RHOL, PL, UL, HL, CSL, VELL, VELTL, WL,
|
| 435 |
& RHOR, PR, UR, HR, CSR, VELR, VELTR, WR
|
| 436 |
|
| 437 |
C ---------
|
| 438 |
C INTERNAL VARIABLES
|
| 439 |
C ---------
|
| 440 |
|
| 441 |
DOUBLEPRECISION A, B, C, D, E, EPS, FA, FB, FC, TOL1,
|
| 442 |
& XM, P, Q, R, S
|
| 443 |
|
| 444 |
C -------------
|
| 445 |
C COMPUTE MACHINE PRECISION
|
| 446 |
C -------------
|
| 447 |
|
| 448 |
EPS = 1.D0
|
| 449 |
10 EPS = EPS/2.D0
|
| 450 |
TOL1 = 1.D0 + EPS
|
| 451 |
IF( TOL1 .GT. 1.D0 ) GO TO 10
|
| 452 |
|
| 453 |
C -------
|
| 454 |
C INITIALIZATION
|
| 455 |
C -------
|
| 456 |
|
| 457 |
A = PMIN
|
| 458 |
B = PMAX
|
| 459 |
CALL GETDVEL2(A,FA)
|
| 460 |
CALL GETDVEL2(B,FB)
|
| 461 |
|
| 462 |
C -----
|
| 463 |
C BEGIN STEP
|
| 464 |
C -----
|
| 465 |
|
| 466 |
20 C = A
|
| 467 |
FC = FA
|
| 468 |
D = B - A
|
| 469 |
E = D
|
| 470 |
30 IF( DABS(FC) .GE. DABS(FB) )GO TO 40
|
| 471 |
A = B
|
| 472 |
B = C
|
| 473 |
C = A
|
| 474 |
FA = FB
|
| 475 |
FB = FC
|
| 476 |
FC = FA
|
| 477 |
|
| 478 |
C --------
|
| 479 |
C CONVERGENCE TEST
|
| 480 |
C --------
|
| 481 |
|
| 482 |
40 TOL1 = 2.D0*EPS*DABS(B) + 0.5D0*TOL
|
| 483 |
XM = 0.5D0*(C - B)
|
| 484 |
IF( DABS(XM) .LE. TOL1 ) GO TO 90
|
| 485 |
IF( FB .EQ. 0.D0 ) GO TO 90
|
| 486 |
|
| 487 |
C ------------
|
| 488 |
C IS BISECTION NECESSARY?
|
| 489 |
C ------------
|
| 490 |
|
| 491 |
IF( DABS(E) .LT. TOL1 ) GO TO 70
|
| 492 |
IF( DABS(FA) .LE. DABS(FB) ) GO TO 70
|
| 493 |
|
| 494 |
C ------------------
|
| 495 |
C IS QUADRATIC INTERPOLATION POSSIBLE?
|
| 496 |
C ------------------
|
| 497 |
|
| 498 |
IF( A .NE. C ) GO TO 50
|
| 499 |
|
| 500 |
C ----------
|
| 501 |
C LINEAR INTERPOLATION
|
| 502 |
C ----------
|
| 503 |
|
| 504 |
S = FB/FA
|
| 505 |
P = 2.D0*XM*S
|
| 506 |
Q = 1.D0 - S
|
| 507 |
GO TO 60
|
| 508 |
|
| 509 |
C ----------------
|
| 510 |
C INVERSE QUADRATIC INTERPOLATION
|
| 511 |
C ----------------
|
| 512 |
|
| 513 |
50 Q = FA/FC
|
| 514 |
R = FB/FC
|
| 515 |
S = FB/FA
|
| 516 |
P = S*(2.D0*XM*Q*(Q - R) - (B - A)*(R - 1.D0))
|
| 517 |
Q = (Q - 1.D0)*(R - 1.D0)*(S - 1.D0)
|
| 518 |
|
| 519 |
C ------
|
| 520 |
C ADJUST SIGNS
|
| 521 |
C ------
|
| 522 |
|
| 523 |
60 IF( P .GT. 0.D0 ) Q = -Q
|
| 524 |
P = DABS(P)
|
| 525 |
|
| 526 |
C --------------
|
| 527 |
C IS INTERPOLATION ACCEPTABLE?
|
| 528 |
C --------------
|
| 529 |
|
| 530 |
IF( (2.D0*P) .GE. (3.D0*XM*Q-DABS(TOL1*Q)) ) GO TO 70
|
| 531 |
IF( P .GE. DABS(0.5D0*E*Q) ) GO TO 70
|
| 532 |
E = D
|
| 533 |
D = P/Q
|
| 534 |
GO TO 80
|
| 535 |
|
| 536 |
C -----
|
| 537 |
C BISECTION
|
| 538 |
C -----
|
| 539 |
|
| 540 |
70 D = XM
|
| 541 |
E = D
|
| 542 |
|
| 543 |
C -------
|
| 544 |
C COMPLETE STEP
|
| 545 |
C -------
|
| 546 |
|
| 547 |
80 A = B
|
| 548 |
FA = FB
|
| 549 |
IF( DABS(D) .GT. TOL1 ) B = B+D
|
| 550 |
IF( DABS(D) .LE. TOL1 ) B = B+DSIGN(TOL1,XM)
|
| 551 |
CALL GETDVEL2(B,FB)
|
| 552 |
IF( (FB*(FC/DABS(FC))) .GT. 0.D0) GO TO 20
|
| 553 |
GO TO 30
|
| 554 |
|
| 555 |
C --
|
| 556 |
C DONE
|
| 557 |
C --
|
| 558 |
|
| 559 |
90 PS = B
|
| 560 |
|
| 561 |
RETURN
|
| 562 |
END
|
| 563 |
|
| 564 |
C ------
|
| 565 |
CN NAME: F L A M B
|
| 566 |
C ------
|
| 567 |
|
| 568 |
CP PURPOSE:
|
| 569 |
CP COMPUTE THE VALUE OF THE SELF-SIMILARITY VARIABLE INSIDE A RAREFACTION
|
| 570 |
CP CONNECTED TO A SPECIFIED LEFT / RIGHT STATE
|
| 571 |
C
|
| 572 |
|
| 573 |
CC COMMENTS:
|
| 574 |
CC NONE
|
| 575 |
|
| 576 |
SUBROUTINE FLAMB(K, A, P, VEL, S, XI)
|
| 577 |
|
| 578 |
IMPLICIT NONE
|
| 579 |
|
| 580 |
C --------
|
| 581 |
C ARGUMENTS
|
| 582 |
C --------
|
| 583 |
|
| 584 |
DOUBLE PRECISION K, A, P, VEL
|
| 585 |
|
| 586 |
CHARACTER*1 S
|
| 587 |
|
| 588 |
DOUBLE PRECISION XI
|
| 589 |
|
| 590 |
C -------
|
| 591 |
C COMMON BLOCKS
|
| 592 |
C -------
|
| 593 |
|
| 594 |
DOUBLE PRECISION G
|
| 595 |
COMMON /GB/ G
|
| 596 |
|
| 597 |
C --------------
|
| 598 |
C INTERNAL VARIABLES
|
| 599 |
C --------------
|
| 600 |
|
| 601 |
DOUBLE PRECISION SIGN
|
| 602 |
DOUBLE PRECISION RHO, H, CS2, VELT2, V2, BETA, DISC
|
| 603 |
|
| 604 |
IF (S.EQ.'L') SIGN = -1.D0
|
| 605 |
|
| 606 |
IF (S.EQ.'R') SIGN = 1.D0
|
| 607 |
|
| 608 |
RHO = (P/K)**(1.D0/G)
|
| 609 |
CS2 = G*(G - 1.D0)*P/(G*P +(G - 1.D0)*RHO)
|
| 610 |
H = 1.D0/(1.D0 - CS2/(G - 1.D0))
|
| 611 |
|
| 612 |
VELT2 = (1.D0 - VEL*VEL)*A*A/(H*H + A*A)
|
| 613 |
V2 = VELT2 + VEL*VEL
|
| 614 |
|
| 615 |
BETA = (1.D0 - V2)*CS2/(1.D0 - CS2)
|
| 616 |
DISC = DSQRT(BETA*(1.D0 + BETA - VEL*VEL))
|
| 617 |
|
| 618 |
XI = (VEL + SIGN*DISC)/(1.D0 + BETA)
|
| 619 |
|
| 620 |
RETURN
|
| 621 |
END
|
| 622 |
|
| 623 |
C --------
|
| 624 |
CN NAME: R A R E F 2
|
| 625 |
C --------
|
| 626 |
|
| 627 |
CP PURPOSE:
|
| 628 |
CP COMPUTE THE FLOW STATE IN A RAREFACTION FOR GIVEN PRE-WAVE STATE
|
| 629 |
C
|
| 630 |
|
| 631 |
CC COMMENTS:
|
| 632 |
CC THE VELOCITY IN THE RAREFACTION IS WRITTEN IN TERMS OF THE PRESCRIBED
|
| 633 |
CC LEFT / RIGHT STATE AND PRESSURE ACCORDING TO EXPRESSIONS (3.25) AND
|
| 634 |
CC (3.26) OF REZZOLLA, ZANOTTI AND PONS, JFM, 2002.
|
| 635 |
CC THE INTEGRAL IN THE VELOCITY EXPRESSION IS COMPUTED THROUGH A GAUSSIAN
|
| 636 |
CC QUADRATURE
|
| 637 |
|
| 638 |
SUBROUTINE RAREF2(XI, PS, RHOA, PA, UA, CSA, VELA, VELTA, S,
|
| 639 |
& RHO, P, U, VEL, VELT)
|
| 640 |
|
| 641 |
IMPLICIT NONE
|
| 642 |
|
| 643 |
INCLUDE 'npoints'
|
| 644 |
|
| 645 |
C ------
|
| 646 |
C ARGUMENTS
|
| 647 |
C ------
|
| 648 |
|
| 649 |
DOUBLE PRECISION XI, PS, RHOA, PA, UA, CSA, VELA, VELTA
|
| 650 |
|
| 651 |
CHARACTER*1 S
|
| 652 |
|
| 653 |
DOUBLE PRECISION RHO, P, U, VEL, VELT
|
| 654 |
|
| 655 |
C ------
|
| 656 |
C COMMON BLOCKS
|
| 657 |
C ------
|
| 658 |
|
| 659 |
DOUBLE PRECISION GB
|
| 660 |
COMMON /GB/ GB
|
| 661 |
|
| 662 |
DOUBLE PRECISION QX(NG), QW(NG)
|
| 663 |
COMMON /INT/ QX, QW
|
| 664 |
|
| 665 |
C --------
|
| 666 |
C INTERNAL VARIABLES
|
| 667 |
C --------
|
| 668 |
|
| 669 |
INTEGER I
|
| 670 |
|
| 671 |
DOUBLE PRECISION HA, WA, SIGN
|
| 672 |
|
| 673 |
DOUBLE PRECISION CONST, K, XIO, XIP, PO, H
|
| 674 |
|
| 675 |
DOUBLE PRECISION SUMW, DIFW, INTEGRAL, XX, RRHO, CCS2, HH,
|
| 676 |
& FUNR, A, FP, DFDP
|
| 677 |
|
| 678 |
HA = 1.D0 + UA + PA/RHOA
|
| 679 |
|
| 680 |
WA = 1.D0/DSQRT(1.D0 - VELA*VELA - VELTA*VELTA)
|
| 681 |
|
| 682 |
CONST = HA*WA*VELTA
|
| 683 |
|
| 684 |
K = PA/RHOA**GB
|
| 685 |
|
| 686 |
IF (S.EQ.'L') SIGN = -1.D0
|
| 687 |
|
| 688 |
IF (S.EQ.'R') SIGN = 1.D0
|
| 689 |
|
| 690 |
CALL FLAMB(K, CONST, PA, VELA, S, XIO)
|
| 691 |
|
| 692 |
PO = PA
|
| 693 |
P = 0.95D0*PA
|
| 694 |
|
| 695 |
20 CONTINUE
|
| 696 |
|
| 697 |
SUMW = 0.5D0*(P + PA)
|
| 698 |
DIFW = 0.5D0*(P - PA)
|
| 699 |
INTEGRAL = 0.D0
|
| 700 |
|
| 701 |
DO 10 I = 1, NG
|
| 702 |
|
| 703 |
XX = DIFW*QX(I) + SUMW
|
| 704 |
RRHO = (XX/K)**(1.D0/GB)
|
| 705 |
CCS2 = GB*(GB - 1.D0)*XX/(GB*XX + (GB - 1.0)*RRHO)
|
| 706 |
HH = 1.D0/(1.D0 - CCS2/(GB - 1.D0))
|
| 707 |
|
| 708 |
FUNR = DSQRT(HH*HH + CONST*CONST*(1.D0 - CCS2))/
|
| 709 |
& (HH*HH + CONST*CONST)/(RRHO*DSQRT(CCS2))
|
| 710 |
|
| 711 |
INTEGRAL = INTEGRAL + DIFW*QW(I)*FUNR
|
| 712 |
|
| 713 |
10 CONTINUE
|
| 714 |
|
| 715 |
A = SIGN*INTEGRAL + 0.5D0*DLOG((1.D0 + VELA)/(1.D0 - VELA))
|
| 716 |
VEL = DTANH(A)
|
| 717 |
|
| 718 |
CALL FLAMB(K, CONST, P, VEL, S, XIP)
|
| 719 |
|
| 720 |
FP = XIP - XI
|
| 721 |
|
| 722 |
DFDP = (XIP - XIO)/(P - PO)
|
| 723 |
|
| 724 |
PO = P
|
| 725 |
XIO = XIP
|
| 726 |
|
| 727 |
P = P - FP/DFDP
|
| 728 |
P = DMAX1(P,PS)
|
| 729 |
|
| 730 |
IF (DABS(FP).GT.1.D-10) GOTO 20
|
| 731 |
|
| 732 |
RHO = (P/K)**(1.D0/GB)
|
| 733 |
|
| 734 |
U = P/RHO/(GB -1.D0)
|
| 735 |
|
| 736 |
H = 1.D0 + U + P/RHO
|
| 737 |
|
| 738 |
VELT = CONST*DSQRT((1.D0 - VEL*VEL)/(CONST*CONST + H*H))
|
| 739 |
|
| 740 |
RETURN
|
| 741 |
END
|
| 742 |
|
| 743 |
C ---------
|
| 744 |
CN NAME: G E T V E L 2
|
| 745 |
C ---------
|
| 746 |
|
| 747 |
CP PURPOSE:
|
| 748 |
CP COMPUTE THE FLOW VELOCITY BEHIND A RAREFACTION OR SHOCK IN TERMS OF THE
|
| 749 |
CP POST-WAVE PRESSURE FOR A GIVEN STATE AHEAD THE WAVE IN A RELATIVISTIC
|
| 750 |
CP FLOW
|
| 751 |
C
|
| 752 |
|
| 753 |
CC COMMENTS:
|
| 754 |
CC THIS ROUTINE CLOSELY FOLLOWS THE EXPRESSIONS IN PONS, MARTI AND MUELLER,
|
| 755 |
CC JFM, 2002
|
| 756 |
|
| 757 |
SUBROUTINE GETVEL2(P, RHOA, PA, UA, HA, CSA, VELA, VELTA, WA, S,
|
| 758 |
& RHO, U, H, CS, VEL, VELT, VSHOCK)
|
| 759 |
|
| 760 |
IMPLICIT NONE
|
| 761 |
|
| 762 |
INCLUDE 'npoints'
|
| 763 |
|
| 764 |
C --------
|
| 765 |
C ARGUMENTS
|
| 766 |
C --------
|
| 767 |
|
| 768 |
DOUBLE PRECISION P, RHOA, PA, UA, HA, CSA, VELA, VELTA, WA
|
| 769 |
CHARACTER*1 S
|
| 770 |
DOUBLE PRECISION RHO, U, H, CS, VEL, VELT, VSHOCK
|
| 771 |
|
| 772 |
C -----
|
| 773 |
C COMMON BLOCKS
|
| 774 |
C -----
|
| 775 |
|
| 776 |
DOUBLE PRECISION GB
|
| 777 |
COMMON /GB/ GB
|
| 778 |
|
| 779 |
DOUBLE PRECISION QX(NG), QW(NG)
|
| 780 |
COMMON /INT/ QX, QW
|
| 781 |
|
| 782 |
C --------
|
| 783 |
C INTERNAL VARIABLES
|
| 784 |
C --------
|
| 785 |
|
| 786 |
INTEGER I
|
| 787 |
|
| 788 |
DOUBLE PRECISION A, B, C, SIGN
|
| 789 |
DOUBLE PRECISION J, WSHOCK
|
| 790 |
DOUBLE PRECISION K, CONST, SUMW, DIFW, INTEGRAL, XX, RRHO,
|
| 791 |
& HH, CCS2, FUNR
|
| 792 |
|
| 793 |
C ------------
|
| 794 |
C LEFT OR RIGHT PROPAGATING WAVE
|
| 795 |
C ------------
|
| 796 |
|
| 797 |
IF (S.EQ.'L') SIGN = -1.D0
|
| 798 |
|
| 799 |
IF (S.EQ.'R') SIGN = 1.D0
|
| 800 |
|
| 801 |
IF (P.GE.PA) THEN
|
| 802 |
|
| 803 |
C ---
|
| 804 |
C SHOCK
|
| 805 |
C ---
|
| 806 |
|
| 807 |
A = 1.D0 - (GB - 1.D0)*(P - PA)/GB/P
|
| 808 |
B = 1.D0 - A
|
| 809 |
C = HA*(PA - P)/RHOA - HA**2
|
| 810 |
|
| 811 |
C ----------------
|
| 812 |
C CHECK FOR UNPHYSICAL ENTHALPIES
|
| 813 |
C ----------------
|
| 814 |
|
| 815 |
IF (C.GT.(B**2/4.D0/A)) STOP
|
| 816 |
& 'GETVEL2: UNPHYSICAL SPECIFIC ENTHALPY IN INTERMEDIATE STATE'
|
| 817 |
|
| 818 |
C ---------------------------------
|
| 819 |
C SPECIFIC ENTHALPY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 820 |
C (OBTAINED FROM THE EQUATION OF STATE AND THE TAUB ADIABAT)
|
| 821 |
C ---------------------------------
|
| 822 |
|
| 823 |
H = (-B + DSQRT(B**2 - 4.D0*A*C))/2.D0/A
|
| 824 |
|
| 825 |
C -----------------------------------
|
| 826 |
C DENSITY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 827 |
C (OBTAINED FROM SPECIFIC ENTHALPY AND THE EQUATION OF STATE)
|
| 828 |
C -----------------------------------
|
| 829 |
|
| 830 |
RHO = GB*P/(GB - 1.D0)/(H - 1.D0)
|
| 831 |
|
| 832 |
C ----------------------------------
|
| 833 |
C SPECIFIC INT. ENERGY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 834 |
C (OBTAINED FROM THE EQUATION OF STATE)
|
| 835 |
C ----------------------------------
|
| 836 |
|
| 837 |
U = P/(GB - 1.D0)/RHO
|
| 838 |
|
| 839 |
C -----------------------------
|
| 840 |
C MASS FLUX ACROSS LEFT WAVE
|
| 841 |
C (OBTAINED FROM THE RANKINE-HUGONIOT RELATIONS)
|
| 842 |
C -----------------------------
|
| 843 |
|
| 844 |
J = SIGN*DSQRT((P - PA)/(HA/RHOA - H/RHO))
|
| 845 |
|
| 846 |
C ------------------------------
|
| 847 |
C SHOCK VELOCITY
|
| 848 |
C (OBTAINED FROM THE DEFINITION OF MASS FLUX)
|
| 849 |
C ------------------------------
|
| 850 |
|
| 851 |
A = J**2 + (RHOA*WA)**2
|
| 852 |
B = -2.D0*VELA*RHOA**2*WA**2
|
| 853 |
C = (RHOA*WA*VELA)**2 - J**2
|
| 854 |
|
| 855 |
VSHOCK = (-B + SIGN*DSQRT(B*B - 4.D0*A*C))/(2.D0*A)
|
| 856 |
WSHOCK = 1.D0/DSQRT(1.D0 - VSHOCK**2)
|
| 857 |
|
| 858 |
C --------------------------
|
| 859 |
C VELOCITY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 860 |
C --------------------------
|
| 861 |
|
| 862 |
A = WSHOCK*(P - PA)/J + HA*WA*VELA
|
| 863 |
B = HA*WA + (P - PA)*(WSHOCK*VELA/J + 1.D0/RHOA/WA)
|
| 864 |
|
| 865 |
VEL = A/B
|
| 866 |
|
| 867 |
A = HA*WA*VELTA
|
| 868 |
|
| 869 |
VELT = A*DSQRT((1.D0 - VEL*VEL)/(H*H + A*A))
|
| 870 |
|
| 871 |
C -----------------------------
|
| 872 |
C SOUND SPEED AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 873 |
C -----------------------------
|
| 874 |
|
| 875 |
CS = DSQRT(GB*P/RHO/H)
|
| 876 |
|
| 877 |
ELSE
|
| 878 |
|
| 879 |
C ------
|
| 880 |
C RAREFACTION
|
| 881 |
C ------
|
| 882 |
|
| 883 |
CONST = HA*WA*VELTA
|
| 884 |
|
| 885 |
K = PA/RHOA**GB
|
| 886 |
|
| 887 |
C --------------------------
|
| 888 |
C DENSITY AT THE LEFT SIDE OF THE CONTACT DISCONTINUITY
|
| 889 |
C --------------------------
|
| 890 |
|
| 891 |
RHO = (P/K)**(1.D0/GB)
|
| 892 |
|
| 893 |
C ----------------------------------
|
| 894 |
C SPECIFIC INT. ENERGY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 895 |
C (OBTAINED FROM THE EQUATION OF STATE)
|
| 896 |
C ----------------------------------
|
| 897 |
|
| 898 |
U = P/(GB - 1.D0)/RHO
|
| 899 |
H = 1.D0 + GB*U
|
| 900 |
|
| 901 |
C ----------------------------------
|
| 902 |
C SOUND SPEED AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 903 |
C ----------------------------------
|
| 904 |
|
| 905 |
CS = DSQRT(GB*P/(RHO + GB*P/(GB - 1.D0)))
|
| 906 |
|
| 907 |
C ----------------------------------
|
| 908 |
C VELOCITY AT THE LEFT OF THE CONTACT DISCONTINUITY
|
| 909 |
C ----------------------------------
|
| 910 |
|
| 911 |
C ------
|
| 912 |
C INTEGRAL
|
| 913 |
C ------
|
| 914 |
|
| 915 |
SUMW = 0.5D0*(P + PA)
|
| 916 |
|
| 917 |
DIFW = 0.5D0*(P - PA)
|
| 918 |
|
| 919 |
INTEGRAL = 0.D0
|
| 920 |
|
| 921 |
DO 110 I = 1, NG
|
| 922 |
|
| 923 |
XX = DIFW*QX(I) + SUMW
|
| 924 |
RRHO = (XX/K)**(1.D0/GB)
|
| 925 |
CCS2 = GB*(GB - 1.D0)*XX/(GB*XX + (GB - 1.0)*RRHO)
|
| 926 |
HH = 1.D0/(1.D0 - CCS2/(GB - 1.D0))
|
| 927 |
|
| 928 |
FUNR = DSQRT(HH*HH + CONST*CONST*(1.D0 - CCS2))/
|
| 929 |
& (HH*HH + CONST*CONST)/(RRHO*DSQRT(CCS2))
|
| 930 |
|
| 931 |
INTEGRAL = INTEGRAL + DIFW*QW(I)*FUNR
|
| 932 |
|
| 933 |
110 CONTINUE
|
| 934 |
|
| 935 |
A = SIGN*INTEGRAL + 0.5D0*DLOG((1.D0 + VELA)/(1.D0 - VELA))
|
| 936 |
VEL = DTANH(A)
|
| 937 |
VELT = CONST*DSQRT((1.D0 - VEL*VEL)/(CONST**2 + H**2))
|
| 938 |
|
| 939 |
END IF
|
| 940 |
|
| 941 |
RETURN
|
| 942 |
END
|
| 943 |
|
| 944 |
C ------
|
| 945 |
CN NAME: G A U L E G
|
| 946 |
C ------
|
| 947 |
|
| 948 |
CP PURPOSE:
|
| 949 |
CP COMPUTE ABCISSAS AND WEIGHTS FOR GAUSS-LEGENDRE QUADRATURE INTEGRATION
|
| 950 |
C
|
| 951 |
|
| 952 |
CC COMMENTS:
|
| 953 |
CC ADAPTED FROM PRESS ET AL., "NUMERICAL RECIPES", CAMBRIDGE, 1988
|
| 954 |
|
| 955 |
SUBROUTINE GAULEG(X1,X2,X,W,N)
|
| 956 |
|
| 957 |
IMPLICIT NONE
|
| 958 |
|
| 959 |
C --------
|
| 960 |
C ARGUMENTS
|
| 961 |
C --------
|
| 962 |
|
| 963 |
INTEGER N
|
| 964 |
DOUBLE PRECISION X1,X2,X(N),W(N)
|
| 965 |
|
| 966 |
C ---------
|
| 967 |
C INTERNAL VARIABLES
|
| 968 |
C ---------
|
| 969 |
|
| 970 |
DOUBLE PRECISION EPS
|
| 971 |
PARAMETER (EPS=3.D-14)
|
| 972 |
INTEGER I, J, M
|
| 973 |
DOUBLE PRECISION P1, P2, P3, PP, XL, XM, Z, Z1
|
| 974 |
|
| 975 |
M = (N + 1)/2
|
| 976 |
XM = 0.5D0*(X2 + X1)
|
| 977 |
XL = 0.5D0*(X2 - X1)
|
| 978 |
|
| 979 |
DO 12 I = 1, M
|
| 980 |
|
| 981 |
Z = COS(3.141592654D0*(I - .25D0)/(N + .5D0))
|
| 982 |
1 CONTINUE
|
| 983 |
|
| 984 |
P1 = 1.D0
|
| 985 |
P2 = 0.D0
|
| 986 |
|
| 987 |
DO 11 J = 1, N
|
| 988 |
|
| 989 |
P3 = P2
|
| 990 |
P2 = P1
|
| 991 |
P1 = ((2.D0*J - 1.D0)*Z*P2 - (J - 1.D0)*P3)/J
|
| 992 |
11 CONTINUE
|
| 993 |
|
| 994 |
PP = N*(Z*P1 - P2)/(Z*Z - 1.D0)
|
| 995 |
Z1 = Z
|
| 996 |
Z = Z1 - P1/PP
|
| 997 |
|
| 998 |
IF (ABS(Z -Z1).GT.EPS) GOTO 1
|
| 999 |
|
| 1000 |
X(I) = XM - XL*Z
|
| 1001 |
X(N+1-I) = XM + XL*Z
|
| 1002 |
W(I) = 2.D0*XL/((1.D0 - Z*Z)*PP*PP)
|
| 1003 |
W(N+1-I) = W(I)
|
| 1004 |
12 CONTINUE
|
| 1005 |
|
| 1006 |
RETURN
|
| 1007 |
END
|