summaryrefslogtreecommitdiff
path: root/Build/source/utils/asymptote/base/ode.asy
blob: 0b88060e081ff97d587d706f712ed77a3023b4d3 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
real stepfactor=2; // Maximum dynamic step size adjustment factor.

struct coefficients
{
  real[] steps;
  real[] factors;
  real[][] weights;
  real[] highOrderWeights;
  real[] lowOrderWeights;
}

struct RKTableau
{
  int order;
  coefficients a;
  void stepDependence(real h, real c, coefficients a) {}
   
  real pgrow;
  real pshrink;
  bool exponential;

  void operator init(int order, real[][] weights, real[] highOrderWeights,
                     real[] lowOrderWeights=new real[],
                     real[] steps=sequence(new real(int i) {
                         return sum(weights[i]);},weights.length),
                     void stepDependence(real, real, coefficients)=null) {
    this.order=order;
    a.steps=steps;
    a.factors=array(a.steps.length+1,1);
    a.weights=weights;
    a.highOrderWeights=highOrderWeights;
    a.lowOrderWeights=lowOrderWeights;
    if(stepDependence != null) {
      this.stepDependence=stepDependence;
      exponential=true;
    }
    pgrow=(order > 0) ? 1/order : 0;
    pshrink=(order > 1) ? 1/(order-1) : pgrow;
  }
}

real[] Coeff={1,1/2,1/6,1/24,1/120,1/720,1/5040,1/40320,1/362880,1/3628800,
              1/39916800.0,1/479001600.0,1/6227020800.0,1/87178291200.0,
              1/1307674368000.0,1/20922789888000.0,1/355687428096000.0,
              1/6402373705728000.0,1/121645100408832000.0,
              1/2432902008176640000.0,1/51090942171709440000.0,
              1/1124000727777607680000.0};

real phi1(real x) {return x != 0 ? expm1(x)/x : 1;}

real phi2(real x)
{
  real x2=x*x;
  if(fabs(x) > 1) return (exp(x)-x-1)/x2;
  real x3=x2*x;
  real x5=x2*x3;
  if(fabs(x) < 0.1) 
    return Coeff[1]+x*Coeff[2]+x2*Coeff[3]+x3*Coeff[4]+x2*x2*Coeff[5]
      +x5*Coeff[6]+x3*x3*Coeff[7]+x5*x2*Coeff[8]+x5*x3*Coeff[9];
    else {
      real x7=x5*x2;
      real x8=x7*x;
      return Coeff[1]+x*Coeff[2]+x2*Coeff[3]+x3*Coeff[4]+x2*x2*Coeff[5]
        +x5*Coeff[6]+x3*x3*Coeff[7]+x7*Coeff[8]+x8*Coeff[9]
        +x8*x*Coeff[10]+x5*x5*Coeff[11]+x8*x3*Coeff[12]+x7*x5*Coeff[13]+
        x8*x5*Coeff[14]+x7*x7*Coeff[15]+x8*x7*Coeff[16]+x8*x8*Coeff[17];
    }
}

real phi3(real x)
{
  real x2=x*x;
  real x3=x2*x;
  if(fabs(x) > 1.6) return (exp(x)-0.5*x2-x-1)/x3;
  real x5=x2*x3;
  if(fabs(x) < 0.1) 
    return Coeff[2]+x*Coeff[3]+x2*Coeff[4]+x3*Coeff[5]
      +x2*x2*Coeff[6]+x5*Coeff[7]+x3*x3*Coeff[8]+x5*x2*Coeff[9]
      +x5*x3*Coeff[10];
    else {
      real x7=x5*x2;
      real x8=x7*x;
      real x16=x8*x8;
      return Coeff[2]+x*Coeff[3]+x2*Coeff[4]+x3*Coeff[5]
        +x2*x2*Coeff[6]+x5*Coeff[7]+x3*x3*Coeff[8]+x5*x2*Coeff[9]
        +x5*x3*Coeff[10]+x8*x*Coeff[11]
        +x5*x5*Coeff[12]+x8*x3*Coeff[13]+x7*x5*Coeff[14]
        +x8*x5*Coeff[15]+x7*x7*Coeff[16]+x8*x7*Coeff[17]+x16*Coeff[18]
        +x16*x*Coeff[19]+x16*x2*Coeff[20];
    }
}

void expfactors(real x, coefficients a) 
{
  for(int i=0; i < a.steps.length; ++i)
    a.factors[i]=exp(x*a.steps[i]);
  a.factors[a.steps.length]=exp(x);
}
      
// First-Order Euler
RKTableau Euler=RKTableau(1,new real[][],
                          new real[] {1});

// First-Order Exponential Euler
RKTableau E_Euler=RKTableau(1,new real[][], new real[] {1},
                            new void(real h, real c, coefficients a) {
                              real x=-c*h;
                              expfactors(x,a);
                              a.highOrderWeights[0]=phi1(x);
                            });

// Second-Order Runge-Kutta
RKTableau RK2=RKTableau(2,new real[][] {{1/2}},
                        new real[] {0,1}, // 2nd order
                        new real[] {1,0}); // 1st order

// Second-Order Exponential Runge-Kutta
RKTableau E_RK2=RKTableau(2,new real[][] {{1/2}},
                          new real[] {0,1}, // 2nd order
                          new real[] {1,0}, // 1st order
                          new void(real h, real c, coefficients a) {
                            real x=-c*h;
                            expfactors(x,a);
                            a.weights[0][0]=1/2*phi1(x/2);
                            real w=phi1(x);
                            a.highOrderWeights[0]=0;
                            a.highOrderWeights[1]=w;
                            a.lowOrderWeights[0]=w;
                          });

// Second-Order Predictor-Corrector
RKTableau PC=RKTableau(2,new real[][] {{1}},
                       new real[] {1/2,1/2}, // 2nd order
                       new real[] {1,0}); // 1st order

// Second-Order Exponential Predictor-Corrector
RKTableau E_PC=RKTableau(2,new real[][] {{1}},
                         new real[] {1/2,1/2}, // 2nd order
                         new real[] {1,0}, // 1st order
                         new void(real h, real c, coefficients a) {
                           real x=-c*h;
                           expfactors(x,a);
                           real w=phi1(x);
                           a.weights[0][0]=w;
                           a.highOrderWeights[0]=w/2;
                           a.highOrderWeights[1]=w/2;
                           a.lowOrderWeights[0]=w;
                         });

// Third-Order Classical Runge-Kutta
RKTableau RK3=RKTableau(3,new real[][] {{1/2},{-1,2}},
                        new real[] {1/6,2/3,1/6});

// Third-Order Bogacki-Shampine Runge-Kutta
RKTableau RK3BS=RKTableau(3,new real[][] {{1/2},{0,3/4}},
                          new real[] {2/9,1/3,4/9}, // 3rd order
                          new real[] {7/24,1/4,1/3,1/8}); // 2nd order

// Third-Order Exponential Bogacki-Shampine Runge-Kutta
RKTableau E_RK3BS=RKTableau(3,new real[][] {{1/2},{0,3/4}},
                            new real[] {2/9,1/3,4/9}, // 3rd order
                            new real[] {7/24,1/4,1/3,1/8}, // 2nd order
                            new void(real h, real c, coefficients a) {
                              real x=-c*h;
                              expfactors(x,a);
                              real w=phi1(x);
                              real w2=phi2(x);
                              a.weights[0][0]=1/2*phi1(x/2);
                              real a11=9/8*phi2(3/4*x)+3/8*phi2(x/2);
                              a.weights[1][0]=3/4*phi1(3/4*x)-a11;
                              a.weights[1][1]=a11;
                              real a21=1/3*w;
                              real a22=4/3*w2-2/9*w;
                              a.highOrderWeights[0]=w-a21-a22;
                              a.highOrderWeights[1]=a21;
                              a.highOrderWeights[2]=a22;
                              a.lowOrderWeights[0]=w-17/12*w2;
                              a.lowOrderWeights[1]=w2/2;
                              a.lowOrderWeights[2]=2/3*w2;
                              a.lowOrderWeights[3]=w2/4;
                            });

// Fourth-Order Classical Runge-Kutta
RKTableau RK4=RKTableau(4,new real[][] {{1/2},{0,1/2},{0,0,1}},
                        new real[] {1/6,1/3,1/3,1/6});

// Fifth-Order Cash-Karp Runge-Kutta
RKTableau RK5=RKTableau(5,new real[][] {{1/5},
                                        {3/40,9/40},
                                        {3/10,-9/10,6/5},
                                        {-11/54,5/2,-70/27,35/27},
                                        {1631/55296,175/512,575/13824,
                                         44275/110592,253/4096}},
  new real[] {37/378,0,250/621,125/594,
              0,512/1771},  // 5th order
  new real[] {2825/27648,0,18575/48384,13525/55296,
              277/14336,1/4}); // 4th order

// Fifth-Order Fehlberg Runge-Kutta
RKTableau RK5F=RKTableau(5,new real[][] {{1/4},
                                         {3/32,9/32},
                                         {1932/2197,-7200/2197,7296/2197},
                                         {439/216,-8,3680/513,-845/4104},
                                         {-8/27,2,-3544/2565,1859/4104,
                                          -11/40}},
  new real[] {16/135,0,6656/12825,28561/56430,-9/50,2/55}, // 5th order
  new real[] {25/216,0,1408/2565,2197/4104,-1/5,0}); // 4th order

// Fifth-Order Dormand-Prince Runge-Kutta
RKTableau RK5DP=RKTableau(5,new real[][] {{1/5},
                                          {3/40,9/40},
                                          {44/45,-56/15,32/9},
                                          {19372/6561,-25360/2187,64448/6561,
                                           -212/729},
                                          {9017/3168,-355/33,46732/5247,49/176,
                                           -5103/18656}},
  new real[] {35/384,0,500/1113,125/192,-2187/6784,
              11/84}, // 5th order
  new real[] {5179/57600,0,7571/16695,393/640,
              -92097/339200,187/2100,1/40}); // 4th order

real error(real error, real initial, real lowOrder, real norm, real diff) 
{
  if(initial != 0 && lowOrder != initial) {
    static real epsilon=realMin/realEpsilon;
    real denom=max(abs(norm),abs(initial))+epsilon;
    return max(error,max(abs(diff)/denom));
  }
  return error;
}

void report(real old, real h, real t)
{
  write("Time step changed from "+(string) old+" to "+(string) h+" at t="+
        (string) t+".");
}

real adjust(real h, real error, real tolmin, real tolmax, RKTableau tableau)
{
  if(error > tolmax)
    h *= max((tolmin/error)^tableau.pshrink,1/stepfactor);
  else if(error > 0 && error < tolmin)
    h *= min((tolmin/error)^tableau.pgrow,stepfactor);
  return h;
}

struct solution
{
  real[] t;
  real[] y;
}

void write(solution S) 
{
  for(int i=0; i < S.t.length; ++i)
    write(S.t[i],S.y[i]);
}

// Integrate dy/dt+cy=f(t,y) from a to b using initial conditions y,
// specifying either the step size h or the number of steps n.
solution integrate(real y, real c=0, real g(real t, real y), real a, real b=a,
                   real h=0, int n=0, bool dynamic=false, real tolmin=0,
                   real tolmax=0, real dtmin=0, real dtmax=realMax,
                   RKTableau tableau, bool verbose=false)
{
  solution S;
  S.t=new real[] {a};
  S.y=new real[]{y};

  if(h == 0) {
    if(b == a) return S;
    if(n == 0) abort("Either n or h must be specified");
    else h=(b-a)/n;
  }

  real f(real t, real y)=(c == 0 || tableau.exponential) ? g :
    new real(real t, real y) {return g(t,y)-c*y;};

  tableau.stepDependence(h,c,tableau.a);
      
  real t=a;
  real f0;
  if(tableau.a.lowOrderWeights.length == 0) dynamic=false;
  bool fsal=dynamic &&
    (tableau.a.lowOrderWeights.length > tableau.a.highOrderWeights.length);
  if(fsal) f0=f(t,y);

  real dt=h;
  while(t < b) {
    h=min(h,b-t);
    if(t+h == t) break;
    if(h != dt) {
      if(verbose) report(dt,h,t);
      tableau.stepDependence(h,c,tableau.a);
      dt=h;
    }
 
    real[] predictions={fsal ? f0 : f(t,y)};
    for(int i=0; i < tableau.a.steps.length; ++i)
      predictions.push(f(t+h*tableau.a.steps[i],
                         tableau.a.factors[i]*y+h*dot(tableau.a.weights[i],
                                                      predictions)));

    real highOrder=h*dot(tableau.a.highOrderWeights,predictions);
    real y0=tableau.a.factors[tableau.a.steps.length]*y;
    if(dynamic) {
      real f1;
      if(fsal) {
        f1=f(t+h,y0+highOrder);
        predictions.push(f1);
      }
      real lowOrder=h*dot(tableau.a.lowOrderWeights,predictions);
      real error;
      error=error(error,y,y0+lowOrder,y0+highOrder,highOrder-lowOrder);
      h=adjust(h,error,tolmin,tolmax,tableau);
      if(h >= dt) {
        t += dt;
        y=y0+highOrder;
        S.t.push(t);
        S.y.push(y);
        f0=f1;
      }
      h=min(max(h,dtmin),dtmax);
    } else {
      t += h;
      y=y0+highOrder;
      S.y.push(y);
    }
  }
  return S;
}

struct Solution
{
  real[] t;
  real[][] y;
}

void write(Solution S) 
{
  for(int i=0; i < S.t.length; ++i) {
    write(S.t[i],tab);
    for(real y : S.y[i])
      write(y,tab);
    write();
  }
}

// Integrate a set of equations, dy/dt=f(t,y), from a to b using initial
// conditions y, specifying either the step size h or the number of steps n.
Solution integrate(real[] y, real[] f(real t, real[] y), real a, real b=a,
                   real h=0, int n=0, bool dynamic=false,
                   real tolmin=0, real tolmax=0, real dtmin=0,
                   real dtmax=realMax, RKTableau tableau, bool verbose=false)
{
  Solution S;
  S.t=new real[] {a};
  S.y=new real[][] {copy(y)};
      
  if(h == 0) {
    if(b == a) return S;
    if(n == 0) abort("Either n or h must be specified");
    else h=(b-a)/n;
  }
  real t=a;
  real[] f0;
  if(tableau.a.lowOrderWeights.length == 0) dynamic=false;
  bool fsal=dynamic &&
    (tableau.a.lowOrderWeights.length > tableau.a.highOrderWeights.length);
  if(fsal) f0=f(t,y);

  real dt=h;
  while(t < b) {
    h=min(h,b-t);
    if(t+h == t) break;
    if(h != dt) {
      if(verbose) report(dt,h,t);
      dt=h;
    }

    real[][] predictions={fsal ? f0 : f(t,y)};
    for(int i=0; i < tableau.a.steps.length; ++i)
      predictions.push(f(t+h*tableau.a.steps[i],
                         y+h*tableau.a.weights[i]*predictions));

    real[] highOrder=h*tableau.a.highOrderWeights*predictions;
    if(dynamic) {
      real[] f1;
      if(fsal) {
        f1=f(t+h,y+highOrder);
        predictions.push(f1);
      }
      real[] lowOrder=h*tableau.a.lowOrderWeights*predictions;
      real error;
      for(int i=0; i < y.length; ++i)
        error=error(error,y[i],y[i]+lowOrder[i],y[i]+highOrder[i],
                    highOrder[i]-lowOrder[i]);
      h=adjust(h,error,tolmin,tolmax,tableau);
      if(h >= dt) {
        t += dt;
        y += highOrder;
        S.t.push(t);
        S.y.push(y);
        f0=f1;
      }
      h=min(max(h,dtmin),dtmax);
    } else {
      t += h;
      y += highOrder;
      S.t.push(t);
      S.y.push(y);
    }
  }
  return S;
}

real[][] finiteDifferenceJacobian(real[] f(real[]), real[] t,
                                  real[] h=sqrtEpsilon*abs(t))
{
  real[] ft=f(t);
  real[][] J=new real[t.length][ft.length];
  real[] ti=copy(t);
  real tlast=ti[0];
  ti[0] += h[0];
  J[0]=(f(ti)-ft)/h[0];
  for(int i=1; i < t.length; ++i) {
    ti[i-1]=tlast;
    tlast=ti[i];
    ti[i] += h[i];
    J[i]=(f(ti)-ft)/h[i];
  }
  return transpose(J);
}

// Solve simultaneous nonlinear system by Newton's method.
real[] newton(int iterations=100, real[] f(real[]), real[][] jacobian(real[]),
              real[] t)
{
  real[] t=copy(t);
  for(int i=0; i < iterations; ++i)
    t += solve(jacobian(t),-f(t));
  return t;
}

real[] solveBVP(real[] f(real, real[]), real a, real b=a, real h=0, int n=0,
                bool dynamic=false, real tolmin=0, real tolmax=0, real dtmin=0,
                real dtmax=realMax, RKTableau tableau, bool verbose=false,
                real[] initial(real[]), real[] discrepancy(real[]),
                real[] guess, int iterations=100)
{
  real[] g(real[] t) {
    real[][] y=integrate(initial(t),f,a,b,h,n,dynamic,tolmin,tolmax,dtmin,dtmax,
                         tableau,verbose).y;return discrepancy(y[y.length-1]);
  }
  real[][] jacobian(real[] t) {return finiteDifferenceJacobian(g,t);}
  return initial(newton(iterations,g,jacobian,guess));
}