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
|
100 |
|
101 |
C ------
|
102 |
C RIGHT STATE
|
103 |
C ------
|
104 |
|
105 |
WRITE(*,*) ' ---- RIGHT STATE --- '
|
106 |
WRITE(*,*) ' PRESSURE : '
|
107 |
READ (*,*) PR
|
108 |
WRITE(*,*) ' DENSITY : '
|
109 |
READ (*,*) RHOR
|
110 |
WRITE(*,*) ' FLOW VELOCITY-X: '
|
111 |
READ (*,*) VELR
|
112 |
WRITE(*,*) ' FLOW VELOCITY-Y: '
|
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
|