# Coverage for pySDC/implementations/problem_classes/acoustic_helpers/standard_integrators.py: 96%

## 259 statements

, created at 2024-09-09 14:59 +0000

1import math

2from decimal import Decimal, getcontext

4import numpy as np

5import scipy.sparse as sp

6import scipy.sparse.linalg as LA

9#

10# Runge-Kutta IMEX methods of order 1 to 3

11#

12class rk_imex:

13 def __init__(self, M_fast, M_slow, order):

14 assert np.shape(M_fast)[0] == np.shape(M_fast)[1], "A_fast must be square"

15 assert np.shape(M_slow)[0] == np.shape(M_slow)[1], "A_slow must be square"

16 assert np.shape(M_fast)[0] == np.shape(M_slow)[0], "A_fast and A_slow must be of the same size"

18 assert order in [1, 2, 3, 4, 5], "Order must be between 2 and 5"

19 self.order = order

21 if self.order == 2:

22 self.A = np.array([[0, 0], [0, 0.5]])

23 self.A_hat = np.array([[0, 0], [0.5, 0]])

24 self.b = np.array([0, 1])

25 self.b_hat = np.array([0, 1])

26 self.nstages = 2

28 elif self.order == 3:

29 # parameter from Pareschi and Russo, J. Sci. Comp. 2005

30 alpha = 0.24169426078821

31 beta = 0.06042356519705

32 eta = 0.12915286960590

33 self.A_hat = np.array([[0, 0, 0, 0], [0, 0, 0, 0], [0, 1.0, 0, 0], [0, 1.0 / 4.0, 1.0 / 4.0, 0]])

34 self.A = np.array(

35 [

36 [alpha, 0, 0, 0],

37 [-alpha, alpha, 0, 0],

38 [0, 1.0 - alpha, alpha, 0],

39 [beta, eta, 0.5 - beta - eta - alpha, alpha],

40 ]

41 )

42 self.b_hat = np.array([0, 1.0 / 6.0, 1.0 / 6.0, 2.0 / 3.0])

43 self.b = self.b_hat

44 self.nstages = 4

46 elif self.order == 4:

47 self.A_hat = np.array(

48 [

49 [0, 0, 0, 0, 0, 0],

50 [1.0 / 2, 0, 0, 0, 0, 0],

51 [13861.0 / 62500.0, 6889.0 / 62500.0, 0, 0, 0, 0],

52 [

53 -116923316275.0 / 2393684061468.0,

54 -2731218467317.0 / 15368042101831.0,

55 9408046702089.0 / 11113171139209.0,

56 0,

57 0,

58 0,

59 ],

60 [

61 -451086348788.0 / 2902428689909.0,

62 -2682348792572.0 / 7519795681897.0,

63 12662868775082.0 / 11960479115383.0,

64 3355817975965.0 / 11060851509271.0,

65 0,

66 0,

67 ],

68 [

69 647845179188.0 / 3216320057751.0,

70 73281519250.0 / 8382639484533.0,

71 552539513391.0 / 3454668386233.0,

72 3354512671639.0 / 8306763924573.0,

73 4040.0 / 17871.0,

74 0,

75 ],

76 ]

77 )

78 self.A = np.array(

79 [

80 [0, 0, 0, 0, 0, 0],

81 [1.0 / 4, 1.0 / 4, 0, 0, 0, 0],

82 [8611.0 / 62500.0, -1743.0 / 31250.0, 1.0 / 4, 0, 0, 0],

83 [5012029.0 / 34652500.0, -654441.0 / 2922500.0, 174375.0 / 388108.0, 1.0 / 4, 0, 0],

84 [

85 15267082809.0 / 155376265600.0,

86 -71443401.0 / 120774400.0,

87 730878875.0 / 902184768.0,

88 2285395.0 / 8070912.0,

89 1.0 / 4,

90 0,

91 ],

92 [82889.0 / 524892.0, 0, 15625.0 / 83664.0, 69875.0 / 102672.0, -2260.0 / 8211, 1.0 / 4],

93 ]

94 )

95 self.b = np.array([82889.0 / 524892.0, 0, 15625.0 / 83664.0, 69875.0 / 102672.0, -2260.0 / 8211, 1.0 / 4])

96 self.b_hat = np.array(

97 [

98 4586570599.0 / 29645900160.0,

99 0,

100 178811875.0 / 945068544.0,

101 814220225.0 / 1159782912.0,

102 -3700637.0 / 11593932.0,

103 61727.0 / 225920.0,

104 ]

105 )

106 self.nstages = 6

108 elif self.order == 5:

109 # from Kennedy and Carpenter

110 # copied from http://www.mcs.anl.gov/petsc/petsc-3.2/src/ts/impls/arkimex/arkimex.c

111 self.A_hat = np.zeros((8, 8))

112 getcontext().prec = 56

113 self.A_hat[1, 0] = Decimal(41.0) / Decimal(100.0)

114 self.A_hat[2, 0] = Decimal(367902744464.0) / Decimal(2072280473677.0)

115 self.A_hat[2, 1] = Decimal(677623207551.0) / Decimal(8224143866563.0)

116 self.A_hat[3, 0] = Decimal(1268023523408.0) / Decimal(10340822734521.0)

117 self.A_hat[3, 1] = 0.0

118 self.A_hat[3, 2] = Decimal(1029933939417.0) / Decimal(13636558850479.0)

119 self.A_hat[4, 0] = Decimal(14463281900351.0) / Decimal(6315353703477.0)

120 self.A_hat[4, 1] = 0.0

121 self.A_hat[4, 2] = Decimal(66114435211212.0) / Decimal(5879490589093.0)

122 self.A_hat[4, 3] = Decimal(-54053170152839.0) / Decimal(4284798021562.0)

123 self.A_hat[5, 0] = Decimal(14090043504691.0) / Decimal(34967701212078.0)

124 self.A_hat[5, 1] = 0.0

125 self.A_hat[5, 2] = Decimal(15191511035443.0) / Decimal(11219624916014.0)

126 self.A_hat[5, 3] = Decimal(-18461159152457.0) / Decimal(12425892160975.0)

127 self.A_hat[5, 4] = Decimal(-281667163811.0) / Decimal(9011619295870.0)

128 self.A_hat[6, 0] = Decimal(19230459214898.0) / Decimal(13134317526959.0)

129 self.A_hat[6, 1] = 0.0

130 self.A_hat[6, 2] = Decimal(21275331358303.0) / Decimal(2942455364971.0)

131 self.A_hat[6, 3] = Decimal(-38145345988419.0) / Decimal(4862620318723.0)

132 self.A_hat[6, 4] = Decimal(-1.0) / Decimal(8.0)

133 self.A_hat[6, 5] = Decimal(-1.0) / Decimal(8.0)

134 self.A_hat[7, 0] = Decimal(-19977161125411.0) / Decimal(11928030595625.0)

135 self.A_hat[7, 1] = 0.0

136 self.A_hat[7, 2] = Decimal(-40795976796054.0) / Decimal(6384907823539.0)

137 self.A_hat[7, 3] = Decimal(177454434618887.0) / Decimal(12078138498510.0)

138 self.A_hat[7, 4] = Decimal(782672205425.0) / Decimal(8267701900261.0)

139 self.A_hat[7, 5] = Decimal(-69563011059811.0) / Decimal(9646580694205.0)

140 self.A_hat[7, 6] = Decimal(7356628210526.0) / Decimal(4942186776405.0)

142 self.b_hat = np.zeros(8)

143 self.b_hat[0] = Decimal(-872700587467.0) / Decimal(9133579230613.0)

144 self.b_hat[1] = 0.0

145 self.b_hat[2] = 0.0

146 self.b_hat[3] = Decimal(22348218063261.0) / Decimal(9555858737531.0)

147 self.b_hat[4] = Decimal(-1143369518992.0) / Decimal(8141816002931.0)

148 self.b_hat[5] = Decimal(-39379526789629.0) / Decimal(19018526304540.0)

149 self.b_hat[6] = Decimal(32727382324388.0) / Decimal(42900044865799.0)

150 self.b_hat[7] = Decimal(41.0) / Decimal(200.0)

152 self.A = np.zeros((8, 8))

153 self.A[1, 0] = Decimal(41.0) / Decimal(200.0)

154 self.A[1, 1] = Decimal(41.0) / Decimal(200.0)

155 self.A[2, 0] = Decimal(41.0) / Decimal(400.0)

156 self.A[2, 1] = Decimal(-567603406766.0) / Decimal(11931857230679.0)

157 self.A[2, 2] = Decimal(41.0) / Decimal(200.0)

158 self.A[3, 0] = Decimal(683785636431.0) / Decimal(9252920307686.0)

159 self.A[3, 1] = 0.0

160 self.A[3, 2] = Decimal(-110385047103.0) / Decimal(1367015193373.0)

161 self.A[3, 3] = Decimal(41.0) / Decimal(200.0)

162 self.A[4, 0] = Decimal(3016520224154.0) / Decimal(10081342136671.0)

163 self.A[4, 1] = 0.0

164 self.A[4, 2] = Decimal(30586259806659.0) / Decimal(12414158314087.0)

165 self.A[4, 3] = Decimal(-22760509404356.0) / Decimal(11113319521817.0)

166 self.A[4, 4] = Decimal(41.0) / Decimal(200.0)

167 self.A[5, 0] = Decimal(218866479029.0) / Decimal(1489978393911.0)

168 self.A[5, 1] = 0.0

169 self.A[5, 2] = Decimal(638256894668.0) / Decimal(5436446318841.0)

170 self.A[5, 3] = Decimal(-1179710474555.0) / Decimal(5321154724896.0)

171 self.A[5, 4] = Decimal(-60928119172.0) / Decimal(8023461067671.0)

172 self.A[5, 5] = Decimal(41.0) / Decimal(200.0)

173 self.A[6, 0] = Decimal(1020004230633.0) / Decimal(5715676835656.0)

174 self.A[6, 1] = 0.0

175 self.A[6, 2] = Decimal(25762820946817.0) / Decimal(25263940353407.0)

176 self.A[6, 3] = Decimal(-2161375909145.0) / Decimal(9755907335909.0)

177 self.A[6, 4] = Decimal(-211217309593.0) / Decimal(5846859502534.0)

178 self.A[6, 5] = Decimal(-4269925059573.0) / Decimal(7827059040749.0)

179 self.A[6, 6] = Decimal(41.0) / Decimal(200.0)

180 self.A[7, 0] = Decimal(-872700587467.0) / Decimal(9133579230613.0)

181 self.A[7, 1] = 0.0

182 self.A[7, 2] = 0.0

183 self.A[7, 3] = Decimal(22348218063261.0) / Decimal(9555858737531.0)

184 self.A[7, 4] = Decimal(-1143369518992.0) / Decimal(8141816002931.0)

185 self.A[7, 5] = Decimal(-39379526789629.0) / Decimal(19018526304540.0)

186 self.A[7, 6] = Decimal(32727382324388.0) / Decimal(42900044865799.0)

187 self.A[7, 7] = Decimal(41.0) / Decimal(200.0)

189 self.b = np.zeros(8)

191 self.b[0] = Decimal(-975461918565.0) / Decimal(9796059967033.0)

192 self.b[1] = 0.0

193 self.b[2] = 0.0

194 self.b[3] = Decimal(78070527104295.0) / Decimal(32432590147079.0)

195 self.b[4] = Decimal(-548382580838.0) / Decimal(3424219808633.0)

196 self.b[5] = Decimal(-33438840321285.0) / Decimal(15594753105479.0)

197 self.b[6] = Decimal(3629800801594.0) / Decimal(4656183773603.0)

198 self.b[7] = Decimal(4035322873751.0) / Decimal(18575991585200.0)

200 self.nstages = 8

202 self.M_fast = sp.csc_matrix(M_fast)

203 self.M_slow = sp.csc_matrix(M_slow)

204 self.ndof = np.shape(M_fast)[0]

206 self.stages = np.zeros((self.nstages, self.ndof), dtype='complex')

208 def timestep(self, u0, dt):

209 # Solve for stages

210 for i in range(0, self.nstages):

211 # Construct RHS

212 rhs = np.copy(u0)

213 for j in range(0, i):

214 rhs += dt * self.A_hat[i, j] * (self.f_slow(self.stages[j, :])) + dt * self.A[i, j] * (

215 self.f_fast(self.stages[j, :])

216 )

218 # Solve for stage i

219 if self.A[i, i] == 0:

220 # Avoid call to spsolve with identity matrix

221 self.stages[i, :] = np.copy(rhs)

222 else:

223 self.stages[i, :] = self.f_fast_solve(rhs, dt * self.A[i, i])

225 # Update

226 for i in range(0, self.nstages):

227 u0 += dt * self.b_hat[i] * (self.f_slow(self.stages[i, :])) + dt * self.b[i] * (

228 self.f_fast(self.stages[i, :])

229 )

231 return u0

233 def f_slow(self, u):

234 return self.M_slow.dot(u)

236 def f_fast(self, u):

237 return self.M_fast.dot(u)

239 def f_fast_solve(self, rhs, alpha):

240 L = sp.eye(self.ndof) - alpha * self.M_fast

241 return LA.spsolve(L, rhs)

244#

245# Trapezoidal rule

246#

247class trapezoidal:

248 def __init__(self, M, alpha=0.5):

249 assert np.shape(M)[0] == np.shape(M)[1], "Matrix M must be quadratic"

250 self.Ndof = np.shape(M)[0]

251 self.M = M

252 self.alpha = alpha

254 def timestep(self, u0, dt):

255 M_trap = sp.eye(self.Ndof) - self.alpha * dt * self.M

256 B_trap = sp.eye(self.Ndof) + (1.0 - self.alpha) * dt * self.M

257 b = B_trap.dot(u0)

258 return LA.spsolve(M_trap, b)

261#

262# A BDF-2 implicit two-step method

263#

264class bdf2:

265 def __init__(self, M):

266 assert np.shape(M)[0] == np.shape(M)[1], "Matrix M must be quadratic"

267 self.Ndof = np.shape(M)[0]

268 self.M = M

270 def firsttimestep(self, u0, dt):

271 b = u0

272 L = sp.eye(self.Ndof) - dt * self.M

273 return LA.spsolve(L, b)

275 def timestep(self, u0, um1, dt):

276 b = (4.0 / 3.0) * u0 - (1.0 / 3.0) * um1

277 L = sp.eye(self.Ndof) - (2.0 / 3.0) * dt * self.M

278 return LA.spsolve(L, b)

281#

282# A diagonally implicit Runge-Kutta method of order 2, 3 or 4

283#

284class dirk:

285 def __init__(self, M, order):

286 assert np.shape(M)[0] == np.shape(M)[1], "Matrix M must be quadratic"

287 self.Ndof = np.shape(M)[0]

288 self.M = sp.csc_matrix(M)

289 self.order = order

291 assert self.order in [2, 22, 3, 4, 5], 'Order must be 2,22,3,4'

293 if self.order == 2:

294 self.nstages = 1

295 self.A = np.zeros((1, 1))

296 self.A[0, 0] = 0.5

297 self.tau = [0.5]

298 self.b = [1.0]

300 if self.order == 22:

301 self.nstages = 2

302 self.A = np.zeros((2, 2))

303 self.A[0, 0] = 1.0 / 3.0

304 self.A[1, 0] = 1.0 / 2.0

305 self.A[1, 1] = 1.0 / 2.0

307 self.tau = np.zeros(2)

308 self.tau[0] = 1.0 / 3.0

309 self.tau[1] = 1.0

311 self.b = np.zeros(2)

312 self.b[0] = 3.0 / 4.0

313 self.b[1] = 1.0 / 4.0

315 if self.order == 3:

316 self.nstages = 2

317 self.A = np.zeros((2, 2))

318 self.A[0, 0] = 0.5 + 1.0 / (2.0 * math.sqrt(3.0))

319 self.A[1, 0] = -1.0 / math.sqrt(3.0)

320 self.A[1, 1] = self.A[0, 0]

322 self.tau = np.zeros(2)

323 self.tau[0] = 0.5 + 1.0 / (2.0 * math.sqrt(3.0))

324 self.tau[1] = 0.5 - 1.0 / (2.0 * math.sqrt(3.0))

326 self.b = np.zeros(2)

327 self.b[0] = 0.5

328 self.b[1] = 0.5

330 if self.order == 4:

331 self.nstages = 3

332 alpha = 2.0 * math.cos(math.pi / 18.0) / math.sqrt(3.0)

334 self.A = np.zeros((3, 3))

335 self.A[0, 0] = (1.0 + alpha) / 2.0

336 self.A[1, 0] = -alpha / 2.0

337 self.A[1, 1] = self.A[0, 0]

338 self.A[2, 0] = 1.0 + alpha

339 self.A[2, 1] = -(1.0 + 2.0 * alpha)

340 self.A[2, 2] = self.A[0, 0]

342 self.tau = np.zeros(3)

343 self.tau[0] = (1.0 + alpha) / 2.0

344 self.tau[1] = 1.0 / 2.0

345 self.tau[2] = (1.0 - alpha) / 2.0

347 self.b = np.zeros(3)

348 self.b[0] = 1.0 / (6.0 * alpha * alpha)

349 self.b[1] = 1.0 - 1.0 / (3.0 * alpha * alpha)

350 self.b[2] = 1.0 / (6.0 * alpha * alpha)

352 if self.order == 5:

353 self.nstages = 5

354 # From Kennedy, Carpenter "Diagonally Implicit Runge-Kutta Methods for

355 # Ordinary Differential Equations. A Review"

356 self.A = np.zeros((5, 5))

357 self.A[0, 0] = 4024571134387.0 / 14474071345096.0

359 self.A[1, 0] = 9365021263232.0 / 12572342979331.0

360 self.A[1, 1] = self.A[0, 0]

362 self.A[2, 0] = 2144716224527.0 / 9320917548702.0

363 self.A[2, 1] = -397905335951.0 / 4008788611757.0

364 self.A[2, 2] = self.A[0, 0]

366 self.A[3, 0] = -291541413000.0 / 6267936762551.0

367 self.A[3, 1] = 226761949132.0 / 4473940808273.0

368 self.A[3, 2] = -1282248297070.0 / 9697416712681.0

369 self.A[3, 3] = self.A[0, 0]

371 self.A[4, 0] = -2481679516057.0 / 4626464057815.0

372 self.A[4, 1] = -197112422687.0 / 6604378783090.0

373 self.A[4, 2] = 3952887910906.0 / 9713059315593.0

374 self.A[4, 3] = 4906835613583.0 / 8134926921134.0

375 self.A[4, 4] = self.A[0, 0]

377 self.b = np.zeros(5)

378 self.b[0] = -2522702558582.0 / 12162329469185.0

379 self.b[1] = 1018267903655.0 / 12907234417901.0

380 self.b[2] = 4542392826351.0 / 13702606430957.0

381 self.b[3] = 5001116467727.0 / 12224457745473.0

382 self.b[4] = 1509636094297.0 / 3891594770934.0

384 self.stages = np.zeros((self.nstages, self.Ndof), dtype='complex')

386 def timestep(self, u0, dt):

387 uend = u0

388 for i in range(0, self.nstages):

389 b = u0

391 # Compute right hand side for this stage's implicit step

392 for j in range(0, i):

393 b = b + self.A[i, j] * dt * self.f(self.stages[j, :])

395 # Implicit solve for current stage

396 self.stages[i, :] = self.f_solve(b, dt * self.A[i, i])

398 # Add contribution of current stage to final value

399 uend = uend + self.b[i] * dt * self.f(self.stages[i, :])

401 return uend

403 #

404 # Returns f(u) = c*u

405 #

406 def f(self, u):

407 return self.M.dot(u)

409 #

410 # Solves (Id - alpha*c)*u = b for u

411 #

412 def f_solve(self, b, alpha):

413 L = sp.eye(self.Ndof) - alpha * self.M

414 return LA.spsolve(L, b)