Coverage for pySDC/implementations/problem_classes/acoustic_helpers/standard_integrators.py: 96%
259 statements
« prev ^ index » next coverage.py v7.5.0, created at 2024-04-29 09:02 +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)