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24 //
25 // G4BulirschStoer class implementation
26 // Based on bulirsch_stoer.hpp from boost
27 //
28 // Author: Dmitry Sorokin, Google Summer of Code 2016
29 // --------------------------------------------------------------------
30
31 #include "G4BulirschStoer.hh"
32
33 #include "G4FieldUtils.hh"
34
35 namespace
36 {
37 constexpr G4double STEPFAC1 = 0.65;
38 constexpr G4double STEPFAC2 = 0.94;
39 constexpr G4double STEPFAC3 = 0.02;
40 constexpr G4double STEPFAC4 = 4.0;
41 constexpr G4double KFAC1 = 0.8;
42 constexpr G4double KFAC2 = 0.9;
43 constexpr G4double inv_STEPFAC1 = 1.0 / STEPFAC1;
44 constexpr G4double inv_STEPFAC4 = 1.0 / STEPFAC4;
45 } // namespace
46
47 G4BulirschStoer::G4BulirschStoer(G4EquationOfMotion* equation,
48 G4int nvar, G4double eps_rel, G4double max_dt)
49 : fnvar(nvar), m_eps_rel(eps_rel), m_midpoint(equation,nvar), m_max_dt(max_dt)
50 {
51 /* initialize sequence of stage numbers and work */
52
53 for(G4int i = 0; i < m_k_max + 1; ++i)
54 {
55 m_interval_sequence[i] = 2 * (i + 1);
56 if (i == 0)
57 {
58 m_cost[i] = m_interval_sequence[i];
59 }
60 else
61 {
62 m_cost[i] = m_cost[i-1] + m_interval_sequence[i];
63 }
64 for(G4int k = 0; k < i; ++k)
65 {
66 const G4double r = static_cast<G4double>(m_interval_sequence[i])
67 / static_cast<G4double>(m_interval_sequence[k]);
68 m_coeff[i][k] = 1.0 / (r * r - 1.0); // coefficients for extrapolation
69 }
70
71 // crude estimate of optimal order
72 m_current_k_opt = 4;
73
74 // no calculation because log10 might not exist for value_type!
75
76 //const G4double logfact = -log10(std::max(eps_rel, 1.0e-12)) * 0.6 + 0.5;
77 //m_current_k_opt = std::max(1.,
78 // std::min(static_cast<G4double>(m_k_max-1), logfact));
79 }
80 }
81
82 G4BulirschStoer::step_result
83 G4BulirschStoer::try_step( const G4double in[], const G4double dxdt[],
84 G4double& t, G4double out[], G4double& dt)
85 {
86 if(m_max_dt < dt)
87 {
88 // given step size is bigger then max_dt set limit and return fail
89 //
90 dt = m_max_dt;
91 return step_result::fail;
92 }
93
94 if (dt != m_dt_last)
95 {
96 reset(); // step size changed from outside -> reset
97 }
98
99 G4bool reject = true;
100
101 G4double new_h = dt;
102
103 /* m_current_k_opt is the estimated current optimal stage number */
104
105 for(G4int k = 0; k <= m_current_k_opt+1; ++k)
106 {
107 // the stage counts are stored in m_interval_sequence
108 //
109 m_midpoint.SetSteps(m_interval_sequence[k]);
110 if(k == 0)
111 {
112 m_midpoint.DoStep(in, dxdt, out, dt);
113 /* the first step, nothing more to do */
114 }
115 else
116 {
117 m_midpoint.DoStep(in, dxdt, m_table[k-1], dt);
118 extrapolate(k, out);
119 // get error estimate
120 for (G4int i = 0; i < fnvar; ++i)
121 {
122 m_err[i] = out[i] - m_table[0][i];
123 }
124 const G4double error =
125 field_utils::relativeError(out, m_err, dt, m_eps_rel);
126 h_opt[k] = calc_h_opt(dt, error, k);
127 work[k] = static_cast<G4double>(m_cost[k]) / h_opt[k];
128
129 if( (k == m_current_k_opt-1) || m_first) // convergence before k_opt ?
130 {
131 if(error < 1.0)
132 {
133 // convergence
134 reject = false;
135 if( (work[k] < KFAC2 * work[k-1]) || (m_current_k_opt <= 2) )
136 {
137 // leave order as is (except we were in first round)
138 m_current_k_opt = std::min(m_k_max - 1 , std::max(2 , k + 1));
139 new_h = h_opt[k];
140 new_h *= static_cast<G4double>(m_cost[k + 1])
141 / static_cast<G4double>(m_cost[k]);
142 }
143 else
144 {
145 m_current_k_opt = std::min(m_k_max - 1, std::max(2, k));
146 new_h = h_opt[k];
147 }
148 break;
149 }
150 if(should_reject(error , k) && !m_first)
151 {
152 reject = true;
153 new_h = h_opt[k];
154 break;
155 }
156 }
157 if(k == m_current_k_opt) // convergence at k_opt ?
158 {
159 if(error < 1.0)
160 {
161 // convergence
162 reject = false;
163 if(work[k-1] < KFAC2 * work[k])
164 {
165 m_current_k_opt = std::max( 2 , m_current_k_opt-1 );
166 new_h = h_opt[m_current_k_opt];
167 }
168 else if( (work[k] < KFAC2 * work[k-1]) && !m_last_step_rejected )
169 {
170 m_current_k_opt = std::min(m_k_max - 1, m_current_k_opt + 1);
171 new_h = h_opt[k];
172 new_h *= static_cast<G4double>(m_cost[m_current_k_opt])
173 / static_cast<G4double>(m_cost[k]);
174 }
175 else
176 {
177 new_h = h_opt[m_current_k_opt];
178 }
179 break;
180 }
181 if(should_reject(error, k))
182 {
183 reject = true;
184 new_h = h_opt[m_current_k_opt];
185 break;
186 }
187 }
188 if(k == m_current_k_opt + 1) // convergence at k_opt+1 ?
189 {
190 if(error < 1.0) // convergence
191 {
192 reject = false;
193 if(work[k-2] < KFAC2 * work[k-1])
194 {
195 m_current_k_opt = std::max(2, m_current_k_opt - 1);
196 }
197 if((work[k] < KFAC2 * work[m_current_k_opt]) && !m_last_step_rejected)
198 {
199 m_current_k_opt = std::min(m_k_max - 1 , k);
200 }
201 new_h = h_opt[m_current_k_opt];
202 }
203 else
204 {
205 reject = true;
206 new_h = h_opt[m_current_k_opt];
207 }
208 break;
209 }
210 }
211 }
212
213 if(!reject)
214 {
215 t += dt;
216 }
217
218 if(!m_last_step_rejected || new_h < dt)
219 {
220 // limit step size
221 new_h = std::min(m_max_dt, new_h);
222 m_dt_last = new_h;
223 dt = new_h;
224 }
225
226 m_last_step_rejected = reject;
227 m_first = false;
228
229 return reject ? step_result::fail : step_result::success;
230 }
231
232 void G4BulirschStoer::reset()
233 {
234 m_first = true;
235 m_last_step_rejected = false;
236 }
237
238 void G4BulirschStoer::extrapolate(std::size_t k , G4double xest[])
239 {
240 /* polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf
241 * uses the obtained intermediate results to extrapolate to dt->0 */
242
243 for(std::size_t j = k - 1 ; j > 0; --j)
244 {
245 for (G4int i = 0; i < fnvar; ++i)
246 {
247 m_table[j-1][i] = m_table[j][i] * (1. + m_coeff[k][j])
248 - m_table[j-1][i] * m_coeff[k][j];
249 }
250 }
251 for (G4int i = 0; i < fnvar; ++i)
252 {
253 xest[i] = m_table[0][i] * (1. + m_coeff[k][0]) - xest[i] * m_coeff[k][0];
254 }
255 }
256
257 G4double
258 G4BulirschStoer::calc_h_opt(G4double h , G4double error , std::size_t k) const
259 {
260 /* calculates the optimal step size for a given error and stage number */
261
262 const G4double expo = 1.0 / (2 * k + 1);
263 const G4double facmin = std::pow(STEPFAC3, expo);
264 G4double fac;
265
266 G4double facminInv= 1.0 / facmin;
267 if (error == 0.0)
268 {
269 fac = facminInv;
270 }
271 else
272 {
273 fac = STEPFAC2 * std::pow(error * inv_STEPFAC1 , -expo);
274 fac = std::max(facmin * inv_STEPFAC4, std::min( facminInv, fac));
275 }
276
277 return h * fac;
278 }
279
280 //why is not used!!??
281 G4bool G4BulirschStoer::set_k_opt(std::size_t k, G4double& dt)
282 {
283 /* calculates the optimal stage number */
284
285 if(k == 1)
286 {
287 m_current_k_opt = 2;
288 return true;
289 }
290 if( (work[k-1] < KFAC1 * work[k]) || (k == m_k_max) ) // order decrease
291 {
292 m_current_k_opt = (G4int)k - 1;
293 dt = h_opt[ m_current_k_opt ];
294 return true;
295 }
296 if( (work[k] < KFAC2 * work[k-1])
297 || m_last_step_rejected || (k == m_k_max-1) )
298 { // same order - also do this if last step got rejected
299 m_current_k_opt = (G4int)k;
300 dt = h_opt[m_current_k_opt];
301 return true;
302 }
303 else { // order increase - only if last step was not rejected
304 m_current_k_opt = (G4int)k + 1;
305 dt = h_opt[m_current_k_opt - 1] * m_cost[m_current_k_opt]
306 / m_cost[m_current_k_opt - 1];
307 return true;
308 }
309 }
310
311 G4bool G4BulirschStoer::in_convergence_window(G4int k) const
312 {
313 if( (k == m_current_k_opt - 1) && !m_last_step_rejected )
314 {
315 return true; // decrease stepsize only if last step was not rejected
316 }
317 return (k == m_current_k_opt) || (k == m_current_k_opt + 1);
318 }
319
320
321 G4bool G4BulirschStoer::should_reject(G4double error, G4int k) const
322 {
323 if(k == m_current_k_opt - 1)
324 {
325 const auto d = G4double(m_interval_sequence[m_current_k_opt]
326 * m_interval_sequence[m_current_k_opt+1]);
327 const auto e = G4double(m_interval_sequence[0]);
328 const G4double e2 = e*e;
329 // step will fail, criterion 17.3.17 in NR
330 return error * e2 * e2 > d * d; // was return error > dOld * dOld; (where dOld= d/e; )
331 }
332 if(k == m_current_k_opt)
333 {
334 const auto d = G4double(m_interval_sequence[m_current_k_opt]);
335 const auto e = G4double(m_interval_sequence[0]);
336 return error * e * e > d * d; // was return error > dOld * dOld; (where dOld= d/e; )
337 }
338 else
339 {
340 return error > 1.0;
341 }
342 }
343