# Coverage for pySDC/implementations/problem_classes/boussinesq_helpers/standard_integrators.py: 0%

## 394 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

6from scipy.sparse.linalg import gmres

8from pySDC.implementations.problem_classes.Boussinesq_2D_FD_imex import boussinesq_2d_imex

9from pySDC.implementations.problem_classes.boussinesq_helpers.helper_classes import logging, Callback

12#

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

14#

15class rk_imex:

16 def __init__(self, problem, order):

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

18 self.order = order

20 if self.order == 1:

21 self.A = np.array([[0, 0], [0, 1]])

22 self.A_hat = np.array([[0, 0], [1, 0]])

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

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

25 self.nstages = 2

27 elif self.order == 2:

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

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

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

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

32 self.nstages = 2

34 elif self.order == 3:

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

36 alpha = 0.24169426078821

37 beta = 0.06042356519705

38 eta = 0.12915286960590

39 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]])

40 self.A = np.array(

41 [

42 [alpha, 0, 0, 0],

43 [-alpha, alpha, 0, 0],

44 [0, 1.0 - alpha, alpha, 0],

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

46 ]

47 )

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

49 self.b = self.b_hat

50 self.nstages = 4

52 elif self.order == 4:

53 self.A_hat = np.array(

54 [

55 [0, 0, 0, 0, 0, 0],

56 [1.0 / 2, 0, 0, 0, 0, 0],

57 [13861.0 / 62500.0, 6889.0 / 62500.0, 0, 0, 0, 0],

58 [

59 -116923316275.0 / 2393684061468.0,

60 -2731218467317.0 / 15368042101831.0,

61 9408046702089.0 / 11113171139209.0,

62 0,

63 0,

64 0,

65 ],

66 [

67 -451086348788.0 / 2902428689909.0,

68 -2682348792572.0 / 7519795681897.0,

69 12662868775082.0 / 11960479115383.0,

70 3355817975965.0 / 11060851509271.0,

71 0,

72 0,

73 ],

74 [

75 647845179188.0 / 3216320057751.0,

76 73281519250.0 / 8382639484533.0,

77 552539513391.0 / 3454668386233.0,

78 3354512671639.0 / 8306763924573.0,

79 4040.0 / 17871.0,

80 0,

81 ],

82 ]

83 )

84 self.A = np.array(

85 [

86 [0, 0, 0, 0, 0, 0],

87 [1.0 / 4, 1.0 / 4, 0, 0, 0, 0],

88 [8611.0 / 62500.0, -1743.0 / 31250.0, 1.0 / 4, 0, 0, 0],

89 [5012029.0 / 34652500.0, -654441.0 / 2922500.0, 174375.0 / 388108.0, 1.0 / 4, 0, 0],

90 [

91 15267082809.0 / 155376265600.0,

92 -71443401.0 / 120774400.0,

93 730878875.0 / 902184768.0,

94 2285395.0 / 8070912.0,

95 1.0 / 4,

96 0,

97 ],

98 [82889.0 / 524892.0, 0, 15625.0 / 83664.0, 69875.0 / 102672.0, -2260.0 / 8211, 1.0 / 4],

99 ]

100 )

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

102 self.b_hat = np.array(

103 [

104 4586570599.0 / 29645900160.0,

105 0,

106 178811875.0 / 945068544.0,

107 814220225.0 / 1159782912.0,

108 -3700637.0 / 11593932.0,

109 61727.0 / 225920.0,

110 ]

111 )

112 self.nstages = 6

114 elif self.order == 5:

115 # from Kennedy and Carpenter

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

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

118 getcontext().prec = 56

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

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

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

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

123 self.A_hat[3, 1] = 0.0

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

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

126 self.A_hat[4, 1] = 0.0

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

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

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

130 self.A_hat[5, 1] = 0.0

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

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

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

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

135 self.A_hat[6, 1] = 0.0

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

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

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

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

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

141 self.A_hat[7, 1] = 0.0

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

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

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

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

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

148 self.b_hat = np.zeros(8)

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

150 self.b_hat[1] = 0.0

151 self.b_hat[2] = 0.0

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

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

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

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

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

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

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

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

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

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

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

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

165 self.A[3, 1] = 0.0

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

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

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

169 self.A[4, 1] = 0.0

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

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

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

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

174 self.A[5, 1] = 0.0

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

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

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

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

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

180 self.A[6, 1] = 0.0

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

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

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

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

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

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

187 self.A[7, 1] = 0.0

188 self.A[7, 2] = 0.0

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

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

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

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

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

195 self.b = np.zeros(8)

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

198 self.b[1] = 0.0

199 self.b[2] = 0.0

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

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

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

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

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

206 self.nstages = 8

208 self.problem = problem

209 self.ndof = np.shape(problem.M)[0]

210 self.logger = logging()

211 self.stages = np.zeros((self.nstages, self.ndof))

213 def timestep(self, u0, dt):

214 # Solve for stages

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

216 # Construct RHS

217 rhs = np.copy(u0)

218 for j in range(0, i):

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

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

221 )

223 # Solve for stage i

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

225 # Avoid call to spsolve with identity matrix

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

227 else:

228 self.stages[i, :] = self.f_fast_solve(rhs, dt * self.A[i, i], u0)

230 # Update

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

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

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

234 )

236 return u0

238 def f_slow(self, u):

239 return self.problem.D_upwind.dot(u)

241 def f_fast(self, u):

242 return self.problem.M.dot(u)

244 def f_fast_solve(self, rhs, alpha, u0):

245 cb = Callback()

246 sol, info = gmres(

247 self.problem.Id - alpha * self.problem.M,

248 rhs,

249 x0=u0,

250 rtol=self.problem.params.gmres_tol_limit,

251 restart=self.problem.params.gmres_restart,

252 maxiter=self.problem.params.gmres_maxiter,

253 atol=0,

254 callback=cb,

255 )

256 if alpha != 0.0:

258 return sol

261#

262# Trapezoidal rule

263#

264class trapezoidal:

265 def __init__(self, problem, alpha=0.5):

266 assert isinstance(problem, boussinesq_2d_imex), "problem is wrong type of object"

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

268 self.order = 2

269 self.logger = logging()

270 self.problem = problem

271 self.alpha = alpha

273 def timestep(self, u0, dt):

274 B_trap = sp.eye(self.Ndof) + self.alpha * dt * (self.problem.D_upwind + self.problem.M)

275 b = B_trap.dot(u0)

276 return self.f_solve(b, alpha=(1.0 - self.alpha) * dt, u0=u0)

278 #

279 # Returns f(u) = c*u

280 #

281 def f(self, u):

282 return self.problem.D_upwind.dot(u) + self.problem.M.dot(u)

284 #

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

286 #

287 def f_solve(self, b, alpha, u0):

288 cb = Callback()

289 sol, info = gmres(

290 self.problem.Id - alpha * (self.problem.D_upwind + self.problem.M),

291 b,

292 x0=u0,

293 rtol=self.problem.params.gmres_tol_limit,

294 restart=self.problem.params.gmres_restart,

295 maxiter=self.problem.params.gmres_maxiter,

296 atol=0,

297 callback=cb,

298 )

299 if alpha != 0.0:

301 return sol

304#

305# A BDF-2 implicit two-step method

306#

307class bdf2:

308 def __init__(self, problem):

309 assert isinstance(problem, boussinesq_2d_imex), "problem is wrong type of object"

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

311 self.order = 2

312 self.logger = logging()

313 self.problem = problem

315 def firsttimestep(self, u0, dt):

316 return self.f_solve(b=u0, alpha=dt, u0=u0)

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

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

320 return self.f_solve(b=b, alpha=(2.0 / 3.0) * dt, u0=u0)

322 #

323 # Returns f(u) = c*u

324 #

325 def f(self, u):

326 return self.problem.D_upwind.dot(u) + self.problem.M.dot(u)

328 #

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

330 #

331 def f_solve(self, b, alpha, u0):

332 cb = Callback()

333 sol, info = gmres(

334 self.problem.Id - alpha * (self.problem.D_upwind + self.problem.M),

335 b,

336 x0=u0,

337 rtol=self.problem.params.gmres_tol_limit,

338 restart=self.problem.params.gmres_restart,

339 maxiter=self.problem.params.gmres_maxiter,

340 atol=0,

341 callback=cb,

342 )

343 if alpha != 0.0:

345 return sol

348#

349# Split-Explicit method

350#

353class SplitExplicit:

354 def __init__(self, problem, method, pparams):

355 assert isinstance(problem, boussinesq_2d_imex), "problem is wrong type of object"

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

357 self.method = method

358 self.logger = logging()

359 self.problem = problem

360 self.pparams = pparams

361 self.NdofTher = 2 * problem.N[0] * problem.N[1]

362 self.NdofMom = 2 * problem.N[0] * problem.N[1]

364 self.ns = None

366 # print("dx ",problem.h[0])

367 # print("dz ",problem.h[1])

369 assert self.method in ["MIS4_4", "RK3"], 'Method must be MIS4_4'

371 if self.method == 'RK3':

372 self.nstages = 3

373 self.aRunge = np.zeros((4, 4))

374 self.aRunge[0, 0] = 1.0 / 3.0

375 self.aRunge[1, 1] = 1.0 / 2.0

376 self.aRunge[2, 2] = 1.0

377 self.dRunge = np.zeros((4, 4))

378 self.gRunge = np.zeros((4, 4))

379 if self.method == 'MIS4_4':

380 self.nstages = 4

381 self.aRunge = np.zeros((4, 4))

382 self.aRunge[0, 0] = 0.38758444641450318

383 self.aRunge[1, 0] = -2.5318448354142823e-002

384 self.aRunge[1, 1] = 0.38668943087310403

385 self.aRunge[2, 0] = 0.20899983523553325

386 self.aRunge[2, 1] = -0.45856648476371231

387 self.aRunge[2, 2] = 0.43423187573425748

388 self.aRunge[3, 0] = -0.10048822195663100

389 self.aRunge[3, 1] = -0.46186171956333327

390 self.aRunge[3, 2] = 0.83045062122462809

391 self.aRunge[3, 3] = 0.27014914900250392

392 self.dRunge = np.zeros((4, 4))

393 self.dRunge[1, 1] = 0.52349249922385610

394 self.dRunge[2, 1] = 1.1683374366893629

395 self.dRunge[2, 2] = -0.75762080241712637

396 self.dRunge[3, 1] = -3.6477233846797109e-002

397 self.dRunge[3, 2] = 0.56936148730740477

398 self.dRunge[3, 3] = 0.47746263002599681

399 self.gRunge = np.zeros((4, 4))

400 self.gRunge[1, 1] = 0.13145089796226542

401 self.gRunge[2, 1] = -0.36855857648747881

402 self.gRunge[2, 2] = 0.33159232636600550

403 self.gRunge[3, 1] = -6.5767130537473045e-002

404 self.gRunge[3, 2] = 4.0591093109036858e-002

405 self.gRunge[3, 3] = 6.4902111640806712e-002

406 self.dtRunge = np.zeros(self.nstages)

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

408 self.dtRunge[i] = 0

409 temp = 1.0

410 for j in range(0, i + 1):

411 self.dtRunge[i] = self.dtRunge[i] + self.aRunge[i, j]

412 temp = temp - self.dRunge[i, j]

413 self.dRunge[i, 0] = temp

414 for j in range(0, i + 1):

415 self.aRunge[i, j] = self.aRunge[i, j] / self.dtRunge[i]

416 self.gRunge[i, j] = self.gRunge[i, j] / self.dtRunge[i]

418 self.U = np.zeros((self.Ndof, self.nstages + 1))

419 self.F = np.zeros((self.Ndof, self.nstages))

420 self.FSlow = np.zeros(self.Ndof)

421 self.nsMin = 8

422 self.logger.nsmall = 0

424 def NumSmallTimeSteps(self, dx, dz, dt):

425 cs = self.pparams['c_s']

426 ns = dt / (0.9 / np.sqrt(1 / (dx * dx) + 1 / (dz * dz)) / cs)

427 ns = max(np.int(np.ceil(ns)), self.nsMin)

428 return ns

430 def timestep(self, u0, dt):

431 self.U[:, 0] = u0

433 self.ns = self.NumSmallTimeSteps(self.problem.h[0], self.problem.h[1], dt)

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

436 self.F[:, i] = self.f_slow(self.U[:, i])

437 self.FSlow[:] = 0.0

438 for j in range(0, i + 1):

439 self.FSlow += self.aRunge[i, j] * self.F[:, j] + self.gRunge[i, j] / dt * (self.U[:, j] - u0)

440 self.U[:, i + 1] = 0

441 for j in range(0, i + 1):

442 self.U[:, i + 1] += self.dRunge[i, j] * self.U[:, j]

443 nsLoc = np.int(np.ceil(self.ns * self.dtRunge[i]))

444 self.logger.nsmall += nsLoc

445 dtLoc = dt * self.dtRunge[i]

446 dTau = dtLoc / nsLoc

447 self.U[:, i + 1] = self.VerletLin(self.U[:, i + 1], self.FSlow, nsLoc, dTau)

448 u0 = self.U[:, self.nstages]

449 return u0

451 def VerletLin(self, u0, FSlow, ns, dTau):

452 for _ in range(0, ns):

453 u0[0 : self.NdofMom] += dTau * (self.f_fastMom(u0) + FSlow[0 : self.NdofMom])

454 u0[self.NdofMom : self.Ndof] += dTau * (self.f_fastTher(u0) + FSlow[self.NdofMom : self.Ndof])

456 return u0

458 def RK3Lin(self, u0, FSlow, ns, dTau):

459 u = u0

460 for _ in range(0, ns):

461 u = u0 + dTau / 3.0 * (self.f_fast(u) + FSlow)

462 u = u0 + dTau / 2.0 * (self.f_fast(u) + FSlow)

463 u = u0 + dTau * (self.f_fast(u) + FSlow)

464 u0 = u

466 return u0

468 def f_slow(self, u):

469 return self.problem.D_upwind.dot(u)

471 def f_fast(self, u):

472 return self.problem.M.dot(u)

474 def f_fastMom(self, u):

475 return self.problem.M[0 : self.NdofMom, self.NdofMom : self.Ndof].dot(u[self.NdofMom : self.Ndof])

477 def f_fastTher(self, u):

478 return self.problem.M[self.NdofMom : self.Ndof, 0 : self.NdofMom].dot(u[0 : self.NdofMom])

481class dirk:

482 def __init__(self, problem, order):

483 assert isinstance(problem, boussinesq_2d_imex), "problem is wrong type of object"

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

485 self.order = order

486 self.logger = logging()

487 self.problem = problem

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

491 if self.order == 2:

492 self.nstages = 1

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

494 self.A[0, 0] = 0.5

495 self.tau = [0.5]

496 self.b = [1.0]

498 if self.order == 22:

499 self.nstages = 2

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

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

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

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

505 self.tau = np.zeros(2)

506 self.tau[0] = 1.0 / 3.0

507 self.tau[1] = 1.0

509 self.b = np.zeros(2)

510 self.b[0] = 3.0 / 4.0

511 self.b[1] = 1.0 / 4.0

513 if self.order == 3:

514 self.nstages = 2

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

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

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

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

520 self.tau = np.zeros(2)

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

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

524 self.b = np.zeros(2)

525 self.b[0] = 0.5

526 self.b[1] = 0.5

528 if self.order == 4:

529 self.nstages = 3

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

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

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

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

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

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

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

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

540 self.tau = np.zeros(3)

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

542 self.tau[1] = 1.0 / 2.0

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

545 self.b = np.zeros(3)

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

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

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

550 if self.order == 5:

551 self.nstages = 5

552 # From Kennedy, Carpenter "Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations.

553 # A Review"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

575 self.b = np.zeros(5)

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

577 self.b[1] = 1018267903655.0 / 12907234417901.0

578 self.b[2] = 4542392826351.0 / 13702606430957.0

579 self.b[3] = 5001116467727.0 / 12224457745473.0

580 self.b[4] = 1509636094297.0 / 3891594770934.0

582 self.stages = np.zeros((self.nstages, self.Ndof))

584 def timestep(self, u0, dt):

585 uend = u0

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

587 b = u0

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

590 for j in range(0, i):

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

593 # Implicit solve for current stage

594 # if i==0:

595 self.stages[i, :] = self.f_solve(b, dt * self.A[i, i], u0)

596 # else:

597 # self.stages[i,:] = self.f_solve( b, dt*self.A[i,i] , self.stages[i-1,:] )

599 # Add contribution of current stage to final value

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

602 return uend

604 #

605 # Returns f(u) = c*u

606 #

607 def f(self, u):

608 return self.problem.D_upwind.dot(u) + self.problem.M.dot(u)

610 #

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

612 #

613 def f_solve(self, b, alpha, u0):

614 cb = Callback()

615 sol, info = gmres(

616 self.problem.Id - alpha * (self.problem.D_upwind + self.problem.M),

617 b,

618 x0=u0,

619 rtol=self.problem.params.gmres_tol_limit,

620 restart=self.problem.params.gmres_restart,

621 maxiter=self.problem.params.gmres_maxiter,

622 atol=0,

623 callback=cb,

624 )

625 if alpha != 0.0: