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mfem/linalg/ode.cpp
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Tzanio 05125e19d5 Removed a space
2020-03-09 15:36:41 -07:00

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// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
#include "operator.hpp"
#include "ode.hpp"
namespace mfem
{
void ODESolver::Init(TimeDependentOperator &f)
{
this->f = &f;
mem_type = GetMemoryType(f.GetMemoryClass());
}
void ForwardEulerSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
dxdt.SetSize(f->Width(), mem_type);
}
void ForwardEulerSolver::Step(Vector &x, double &t, double &dt)
{
f->SetTime(t);
f->Mult(x, dxdt);
x.Add(dt, dxdt);
t += dt;
}
void RK2Solver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
int n = f->Width();
dxdt.SetSize(n, mem_type);
x1.SetSize(n, mem_type);
}
void RK2Solver::Step(Vector &x, double &t, double &dt)
{
// 0 |
// a | a
// ---+--------
// | 1-b b b = 1/(2a)
const double b = 0.5/a;
f->SetTime(t);
f->Mult(x, dxdt);
add(x, (1. - b)*dt, dxdt, x1);
x.Add(a*dt, dxdt);
f->SetTime(t + a*dt);
f->Mult(x, dxdt);
add(x1, b*dt, dxdt, x);
t += dt;
}
void RK3SSPSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
int n = f->Width();
y.SetSize(n, mem_type);
k.SetSize(n, mem_type);
}
void RK3SSPSolver::Step(Vector &x, double &t, double &dt)
{
// x0 = x, t0 = t, k0 = dt*f(t0, x0)
f->SetTime(t);
f->Mult(x, k);
// x1 = x + k0, t1 = t + dt, k1 = dt*f(t1, x1)
add(x, dt, k, y);
f->SetTime(t + dt);
f->Mult(y, k);
// x2 = 3/4*x + 1/4*(x1 + k1), t2 = t + 1/2*dt, k2 = dt*f(t2, x2)
y.Add(dt, k);
add(3./4, x, 1./4, y, y);
f->SetTime(t + dt/2);
f->Mult(y, k);
// x3 = 1/3*x + 2/3*(x2 + k2), t3 = t + dt
y.Add(dt, k);
add(1./3, x, 2./3, y, x);
t += dt;
}
void RK4Solver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
int n = f->Width();
y.SetSize(n, mem_type);
k.SetSize(n, mem_type);
z.SetSize(n, mem_type);
}
void RK4Solver::Step(Vector &x, double &t, double &dt)
{
// 0 |
// 1/2 | 1/2
// 1/2 | 0 1/2
// 1 | 0 0 1
// -----+-------------------
// | 1/6 1/3 1/3 1/6
f->SetTime(t);
f->Mult(x, k); // k1
add(x, dt/2, k, y);
add(x, dt/6, k, z);
f->SetTime(t + dt/2);
f->Mult(y, k); // k2
add(x, dt/2, k, y);
z.Add(dt/3, k);
f->Mult(y, k); // k3
add(x, dt, k, y);
z.Add(dt/3, k);
f->SetTime(t + dt);
f->Mult(y, k); // k4
add(z, dt/6, k, x);
t += dt;
}
ExplicitRKSolver::ExplicitRKSolver(int _s, const double *_a, const double *_b,
const double *_c)
{
s = _s;
a = _a;
b = _b;
c = _c;
k = new Vector[s];
}
void ExplicitRKSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
int n = f->Width();
y.SetSize(n, mem_type);
for (int i = 0; i < s; i++)
{
k[i].SetSize(n, mem_type);
}
}
void ExplicitRKSolver::Step(Vector &x, double &t, double &dt)
{
// 0 |
// c[0] | a[0]
// c[1] | a[1] a[2]
// ... | ...
// c[s-2] | ... a[s(s-1)/2-1]
// --------+---------------------
// | b[0] b[1] ... b[s-1]
f->SetTime(t);
f->Mult(x, k[0]);
for (int l = 0, i = 1; i < s; i++)
{
add(x, a[l++]*dt, k[0], y);
for (int j = 1; j < i; j++)
{
y.Add(a[l++]*dt, k[j]);
}
f->SetTime(t + c[i-1]*dt);
f->Mult(y, k[i]);
}
for (int i = 0; i < s; i++)
{
x.Add(b[i]*dt, k[i]);
}
t += dt;
}
ExplicitRKSolver::~ExplicitRKSolver()
{
delete [] k;
}
const double RK6Solver::a[] =
{
.6e-1,
.1923996296296296296296296296296296296296e-1,
.7669337037037037037037037037037037037037e-1,
.35975e-1,
0.,
.107925,
1.318683415233148260919747276431735612861,
0.,
-5.042058063628562225427761634715637693344,
4.220674648395413964508014358283902080483,
-41.87259166432751461803757780644346812905,
0.,
159.4325621631374917700365669070346830453,
-122.1192135650100309202516203389242140663,
5.531743066200053768252631238332999150076,
-54.43015693531650433250642051294142461271,
0.,
207.0672513650184644273657173866509835987,
-158.6108137845899991828742424365058599469,
6.991816585950242321992597280791793907096,
-.1859723106220323397765171799549294623692e-1,
-54.66374178728197680241215648050386959351,
0.,
207.9528062553893734515824816699834244238,
-159.2889574744995071508959805871426654216,
7.018743740796944434698170760964252490817,
-.1833878590504572306472782005141738268361e-1,
-.5119484997882099077875432497245168395840e-3
};
const double RK6Solver::b[] =
{
.3438957868357036009278820124728322386520e-1,
0.,
0.,
.2582624555633503404659558098586120858767,
.4209371189673537150642551514069801967032,
4.405396469669310170148836816197095664891,
-176.4831190242986576151740942499002125029,
172.3641334014150730294022582711902413315
};
const double RK6Solver::c[] =
{
.6e-1,
.9593333333333333333333333333333333333333e-1,
.1439,
.4973,
.9725,
.9995,
1.,
};
const double RK8Solver::a[] =
{
.5e-1,
-.69931640625e-2,
.1135556640625,
.399609375e-1,
0.,
.1198828125,
.3613975628004575124052940721184028345129,
0.,
-1.341524066700492771819987788202715834917,
1.370126503900035259414693716084313000404,
.490472027972027972027972027972027972028e-1,
0.,
0.,
.2350972042214404739862988335493427143122,
.180855592981356728810903963653454488485,
.6169289044289044289044289044289044289044e-1,
0.,
0.,
.1123656831464027662262557035130015442303,
-.3885046071451366767049048108111244567456e-1,
.1979188712522045855379188712522045855379e-1,
-1.767630240222326875735597119572145586714,
0.,
0.,
-62.5,
-6.061889377376669100821361459659331999758,
5.650823198222763138561298030600840174201,
65.62169641937623283799566054863063741227,
-1.180945066554970799825116282628297957882,
0.,
0.,
-41.50473441114320841606641502701994225874,
-4.434438319103725011225169229846100211776,
4.260408188586133024812193710744693240761,
43.75364022446171584987676829438379303004,
.787142548991231068744647504422630755086e-2,
-1.281405999441488405459510291182054246266,
0.,
0.,
-45.04713996013986630220754257136007322267,
-4.731362069449576477311464265491282810943,
4.514967016593807841185851584597240996214,
47.44909557172985134869022392235929015114,
.1059228297111661135687393955516542875228e-1,
-.5746842263844616254432318478286296232021e-2,
-1.724470134262485191756709817484481861731,
0.,
0.,
-60.92349008483054016518434619253765246063,
-5.95151837622239245520283276706185486829,
5.556523730698456235979791650843592496839,
63.98301198033305336837536378635995939281,
.1464202825041496159275921391759452676003e-1,
.6460408772358203603621865144977650714892e-1,
-.7930323169008878984024452548693373291447e-1,
-3.301622667747079016353994789790983625569,
0.,
0.,
-118.011272359752508566692330395789886851,
-10.14142238845611248642783916034510897595,
9.139311332232057923544012273556827000619,
123.3759428284042683684847180986501894364,
4.623244378874580474839807625067630924792,
-3.383277738068201923652550971536811240814,
4.527592100324618189451265339351129035325,
-5.828495485811622963193088019162985703755
};
const double RK8Solver::b[] =
{
.4427989419007951074716746668098518862111e-1,
0.,
0.,
0.,
0.,
.3541049391724448744815552028733568354121,
.2479692154956437828667629415370663023884,
-15.69420203883808405099207034271191213468,
25.08406496555856261343930031237186278518,
-31.73836778626027646833156112007297739997,
22.93828327398878395231483560344797018313,
-.2361324633071542145259900641263517600737
};
const double RK8Solver::c[] =
{
.5e-1,
.1065625,
.15984375,
.39,
.465,
.155,
.943,
.901802041735856958259707940678372149956,
.909,
.94,
1.,
};
AdamsBashforthSolver::AdamsBashforthSolver(int _s, const double *_a)
{
s = 0;
smax = std::min(_s,5);
a = _a;
k = new Vector[5];
if (smax <= 2)
{
RKsolver = new RK2Solver();
}
else if (smax == 3)
{
RKsolver = new RK3SSPSolver();
}
else
{
RKsolver = new RK4Solver();
}
}
void AdamsBashforthSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
RKsolver->Init(_f);
idx.SetSize(smax);
for (int i = 0; i < smax; i++)
{
idx[i] = (smax-i)%smax;
k[i].SetSize(f->Width());
}
s = 0;
}
void AdamsBashforthSolver::Step(Vector &x, double &t, double &dt)
{
s++;
s = std::min(s, smax);
if (s == smax)
{
f->SetTime(t);
f->Mult(x, k[idx[0]]);
for (int i = 0; i < s; i++)
{
x.Add(a[i]*dt, k[idx[i]]);
}
}
else
{
f->Mult(x,k[idx[0]]);
RKsolver->Step(x,t,dt);
}
t += dt;
// Shift the index
for (int i = 0; i < smax; i++) { idx[i] = ++idx[i]%smax; }
}
const double AB1Solver::a[] =
{1.0};
const double AB2Solver::a[] =
{1.5,-0.5};
const double AB3Solver::a[] =
{23.0/12.0,-4.0/3.0, 5.0/12.0};
const double AB4Solver::a[] =
{55.0/24.0,-59.0/24.0, 37.0/24.0,-9.0/24.0};
const double AB5Solver::a[] =
{1901.0/720.0,-2774.0/720.0, 2616.0/720.0,-1274.0/720.0, 251.0/720.0};
AdamsMoultonSolver::AdamsMoultonSolver(int _s, const double *_a)
{
s = 0;
smax = std::min(_s+1,5);
a = _a;
k = new Vector[5];
if (smax <= 3)
{
RKsolver = new SDIRK23Solver();
}
else
{
RKsolver = new SDIRK34Solver();
}
}
void AdamsMoultonSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
RKsolver->Init(_f);
int n = f->Width();
idx.SetSize(smax);
for (int i = 0; i < smax; i++)
{
idx[i] = (smax-i)%smax;
k[i].SetSize(n);
}
s = 0;
}
void AdamsMoultonSolver::Step(Vector &x, double &t, double &dt)
{
if ((s == 0)&&(smax>1))
{
f->Mult(x,k[idx[1]]);
}
s++;
s = std::min(s, smax);
if (s >= smax-1)
{
f->SetTime(t);
for (int i = 1; i < smax; i++)
{
x.Add(a[i]*dt, k[idx[i]]);
}
f->ImplicitSolve(a[0]*dt, x, k[idx[0]]);
x.Add(a[0]*dt, k[idx[0]]);
}
else
{
RKsolver->Step(x,t,dt);
f->Mult(x,k[idx[0]]);
}
t += dt;
// Shift the index
for (int i = 0; i < smax; i++) { idx[i] = ++idx[i]%smax; }
}
const double AM0Solver::a[] =
{1.0};
const double AM1Solver::a[] =
{0.5, 0.5};
const double AM2Solver::a[] =
{5.0/12.0, 2.0/3.0, -1.0/12.0};
const double AM3Solver::a[] =
{3.0/8.0, 19.0/24.0,-5.0/24.0, 1.0/24.0};
const double AM4Solver::a[] =
{251.0/720.0,646.0/720.0,-264.0/720.0, 106.0/720.0, -19.0/720.0};
void BackwardEulerSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
}
void BackwardEulerSolver::Step(Vector &x, double &t, double &dt)
{
f->SetTime(t + dt);
f->ImplicitSolve(dt, x, k); // solve for k: k = f(x + dt*k, t + dt)
x.Add(dt, k);
t += dt;
}
void ImplicitMidpointSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
}
void ImplicitMidpointSolver::Step(Vector &x, double &t, double &dt)
{
f->SetTime(t + dt/2);
f->ImplicitSolve(dt/2, x, k);
x.Add(dt, k);
t += dt;
}
SDIRK23Solver::SDIRK23Solver(int gamma_opt)
{
if (gamma_opt == 0)
{
gamma = (3. - sqrt(3.))/6.; // not A-stable, order 3
}
else if (gamma_opt == 2)
{
gamma = (2. - sqrt(2.))/2.; // L-stable, order 2
}
else if (gamma_opt == 3)
{
gamma = (2. + sqrt(2.))/2.; // L-stable, order 2
}
else
{
gamma = (3. + sqrt(3.))/6.; // A-stable, order 3
}
}
void SDIRK23Solver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
y.SetSize(f->Width(), mem_type);
}
void SDIRK23Solver::Step(Vector &x, double &t, double &dt)
{
// with a = gamma:
// a | a
// 1-a | 1-2a a
// ------+-----------
// | 1/2 1/2
// note: with gamma_opt=3, both solve are outside [t,t+dt] since a>1
f->SetTime(t + gamma*dt);
f->ImplicitSolve(gamma*dt, x, k);
add(x, (1.-2.*gamma)*dt, k, y); // y = x + (1-2*gamma)*dt*k
x.Add(dt/2, k);
f->SetTime(t + (1.-gamma)*dt);
f->ImplicitSolve(gamma*dt, y, k);
x.Add(dt/2, k);
t += dt;
}
void SDIRK34Solver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
y.SetSize(f->Width(), mem_type);
z.SetSize(f->Width(), mem_type);
}
void SDIRK34Solver::Step(Vector &x, double &t, double &dt)
{
// a | a
// 1/2 | 1/2-a a
// 1-a | 2a 1-4a a
// ------+--------------------
// | b 1-2b b
// note: two solves are outside [t,t+dt] since c1=a>1, c3=1-a<0
const double a = 1./sqrt(3.)*cos(M_PI/18.) + 0.5;
const double b = 1./(6.*(2.*a-1.)*(2.*a-1.));
f->SetTime(t + a*dt);
f->ImplicitSolve(a*dt, x, k);
add(x, (0.5-a)*dt, k, y);
add(x, (2.*a)*dt, k, z);
x.Add(b*dt, k);
f->SetTime(t + dt/2);
f->ImplicitSolve(a*dt, y, k);
z.Add((1.-4.*a)*dt, k);
x.Add((1.-2.*b)*dt, k);
f->SetTime(t + (1.-a)*dt);
f->ImplicitSolve(a*dt, z, k);
x.Add(b*dt, k);
t += dt;
}
void SDIRK33Solver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
y.SetSize(f->Width(), mem_type);
}
void SDIRK33Solver::Step(Vector &x, double &t, double &dt)
{
// a | a
// c | c-a a
// 1 | b 1-a-b a
// -----+----------------
// | b 1-a-b a
const double a = 0.435866521508458999416019;
const double b = 1.20849664917601007033648;
const double c = 0.717933260754229499708010;
f->SetTime(t + a*dt);
f->ImplicitSolve(a*dt, x, k);
add(x, (c-a)*dt, k, y);
x.Add(b*dt, k);
f->SetTime(t + c*dt);
f->ImplicitSolve(a*dt, y, k);
x.Add((1.-a-b)*dt, k);
f->SetTime(t + dt);
f->ImplicitSolve(a*dt, x, k);
x.Add(a*dt, k);
t += dt;
}
void GeneralizedAlphaSolver::Init(TimeDependentOperator &_f)
{
ODESolver::Init(_f);
k.SetSize(f->Width(), mem_type);
y.SetSize(f->Width(), mem_type);
xdot.SetSize(f->Width(), mem_type);
xdot = 0.0;
first = true;
}
void GeneralizedAlphaSolver::SetRhoInf(double rho_inf)
{
rho_inf = (rho_inf > 1.0) ? 1.0 : rho_inf;
rho_inf = (rho_inf < 0.0) ? 0.0 : rho_inf;
// According to Jansen
alpha_m = 0.5*(3.0 - rho_inf)/(1.0 + rho_inf);
alpha_f = 1.0/(1.0 + rho_inf);
gamma = 0.5 + alpha_m - alpha_f;
}
void GeneralizedAlphaSolver::PrintProperties(std::ostream &out)
{
out << "Generalized alpha time integrator:" << std::endl;
out << "alpha_m = " << alpha_m << std::endl;
out << "alpha_f = " << alpha_f << std::endl;
out << "gamma = " << gamma << std::endl;
if (gamma == 0.5 + alpha_m - alpha_f)
{
out<<"Second order"<<" and ";
}
else
{
out<<"First order"<<" and ";
}
if ((alpha_m >= alpha_f)&&(alpha_f >= 0.5))
{
out<<"Stable"<<std::endl;
}
else
{
out<<"Unstable"<<std::endl;
}
}
// This routine assumes xdot is initialized.
void GeneralizedAlphaSolver::Step(Vector &x, double &t, double &dt)
{
if (first)
{
f->Mult(x,xdot);
first = false;
}
// Set y = x + alpha_f*(1.0 - (gamma/alpha_m))*dt*xdot
add(x, alpha_f*(1.0 - (gamma/alpha_m))*dt, xdot, y);
// Solve k = f(y + dt_eff*k)
double dt_eff = (gamma*alpha_f/alpha_m)*dt;
f->SetTime(t + alpha_f*dt);
f->ImplicitSolve(dt_eff, y, k);
// Update x and xdot
x.Add((1.0 - (gamma/alpha_m))*dt, xdot);
x.Add( (gamma/alpha_m) *dt, k);
xdot *= (1.0-(1.0/alpha_m));
xdot.Add((1.0/alpha_m),k);
t += dt;
}
void
SIASolver::Init(Operator &P, TimeDependentOperator & F)
{
P_ = &P; F_ = &F;
dp_.SetSize(F_->Height());
dq_.SetSize(P_->Height());
}
void
SIA1Solver::Step(Vector &q, Vector &p, double &t, double &dt)
{
F_->SetTime(t);
F_->Mult(q,dp_);
p.Add(dt,dp_);
P_->Mult(p,dq_);
q.Add(dt,dq_);
t += dt;
}
void
SIA2Solver::Step(Vector &q, Vector &p, double &t, double &dt)
{
P_->Mult(p,dq_);
q.Add(0.5*dt,dq_);
F_->SetTime(t+0.5*dt);
F_->Mult(q,dp_);
p.Add(dt,dp_);
P_->Mult(p,dq_);
q.Add(0.5*dt,dq_);
t += dt;
}
SIAVSolver::SIAVSolver(int order)
: order_(order)
{
a_.SetSize(order);
b_.SetSize(order);
switch (order_)
{
case 1:
a_[0] = 1.0;
b_[0] = 1.0;
break;
case 2:
a_[0] = 0.5;
a_[1] = 0.5;
b_[0] = 0.0;
b_[1] = 1.0;
break;
case 3:
a_[0] = 2.0/3.0;
a_[1] = -2.0/3.0;
a_[2] = 1.0;
b_[0] = 7.0/24.0;
b_[1] = 0.75;
b_[2] = -1.0/24.0;
break;
case 4:
a_[0] = (2.0+pow(2.0,1.0/3.0)+pow(2.0,-1.0/3.0))/6.0;
a_[1] = (1.0-pow(2.0,1.0/3.0)-pow(2.0,-1.0/3.0))/6.0;
a_[2] = a_[1];
a_[3] = a_[0];
b_[0] = 0.0;
b_[1] = 1.0/(2.0-pow(2.0,1.0/3.0));
b_[2] = 1.0/(1.0-pow(2.0,2.0/3.0));
b_[3] = b_[1];
break;
default:
MFEM_ASSERT(false, "Unsupported order in SIAVSolver");
};
}
void
SIAVSolver::Step(Vector &q, Vector &p, double &t, double &dt)
{
for (int i=0; i<order_; i++)
{
if ( b_[i] != 0.0 )
{
F_->SetTime(t);
if ( F_->isExplicit() )
{
F_->Mult(q, dp_);
}
else
{
F_->ImplicitSolve(b_[i] * dt, q, dp_);
}
p.Add(b_[i] * dt, dp_);
}
P_->Mult(p, dq_);
q.Add(a_[i] * dt, dq_);
t += a_[i] * dt;
}
}
void SecondOrderODESolver::Init(SecondOrderTimeDependentOperator &f)
{
this->f = &f;
mem_type = GetMemoryType(f.GetMemoryClass());
}
void NewmarkSolver::Init(SecondOrderTimeDependentOperator &_f)
{
SecondOrderODESolver::Init(_f);
d2xdt2.SetSize(f->Width());
d2xdt2 = 0.0;
first = true;
}
void NewmarkSolver::PrintProperties(std::ostream &out)
{
out << "Newmark time integrator:" << std::endl;
out << "beta = " << beta << std::endl;
out << "gamma = " << gamma << std::endl;
if (gamma == 0.5)
{
out<<"Second order"<<" and ";
}
else
{
out<<"First order"<<" and ";
}
if ((gamma >= 0.5) && (beta >= (gamma + 0.5)*(gamma + 0.5)/4))
{
out<<"A-Stable"<<std::endl;
}
else if ((gamma >= 0.5) && (beta >= 0.5*gamma))
{
out<<"Conditionally stable"<<std::endl;
}
else
{
out<<"Unstable"<<std::endl;
}
}
void NewmarkSolver::Step(Vector &x, Vector &dxdt, double &t, double &dt)
{
double fac0 = 0.5 - beta;
double fac2 = 1.0 - gamma;
double fac3 = beta;
double fac4 = gamma;
// In the first pass compute d2xdt2 directy from operator.
if (first)
{
f->Mult(x, dxdt, d2xdt2);
first = false;
}
f->SetTime(t + dt);
x.Add(dt, dxdt);
x.Add(fac0*dt*dt, d2xdt2);
dxdt.Add(fac2*dt, d2xdt2);
f->SetTime(t + dt);
f->ImplicitSolve(fac3*dt*dt, fac4*dt, x, dxdt, d2xdt2);
x .Add(fac3*dt*dt, d2xdt2);
dxdt.Add(fac4*dt, d2xdt2);
t += dt;
}
void GeneralizedAlpha2Solver::Init(SecondOrderTimeDependentOperator &_f)
{
SecondOrderODESolver::Init(_f);
xa.SetSize(f->Width());
va.SetSize(f->Width());
aa.SetSize(f->Width());
d2xdt2.SetSize(f->Width());
d2xdt2 = 0.0;
first = true;
}
void GeneralizedAlpha2Solver::PrintProperties(std::ostream &out)
{
out << "Generalized alpha time integrator:" << std::endl;
out << "alpha_m = " << alpha_m << std::endl;
out << "alpha_f = " << alpha_f << std::endl;
out << "beta = " << beta << std::endl;
out << "gamma = " << gamma << std::endl;
if (gamma == 0.5 + alpha_m - alpha_f)
{
out<<"Second order"<<" and ";
}
else
{
out<<"First order"<<" and ";
}
if ((alpha_m >= alpha_f)&&
(alpha_f >= 0.5) &&
(beta >= 0.25 + 0.5*(alpha_m - alpha_f)))
{
out<<"Stable"<<std::endl;
}
else
{
out<<"Unstable"<<std::endl;
}
}
void GeneralizedAlpha2Solver::Step(Vector &x, Vector &dxdt,
double &t, double &dt)
{
double fac0 = (0.5 - (beta/alpha_m));
double fac1 = alpha_f;
double fac2 = alpha_f*(1.0 - (gamma/alpha_m));
double fac3 = beta*alpha_f/alpha_m;
double fac4 = gamma*alpha_f/alpha_m;
double fac5 = alpha_m;
// In the first pass compute d2xdt2 directy from operator.
if (first)
{
f->Mult(x, dxdt, d2xdt2);
first = false;
}
// Predict alpha levels
add(dxdt, fac0*dt, d2xdt2, va);
add(x, fac1*dt, va, xa);
add(dxdt, fac2*dt, d2xdt2, va);
// Solve alpha levels
f->SetTime(t + dt);
f->ImplicitSolve(fac3*dt*dt, fac4*dt, xa, va, aa);
// Correct alpha levels
xa.Add(fac3*dt*dt, aa);
va.Add(fac4*dt, aa);
// Extrapolate
x *= 1.0 - 1.0/fac1;
x.Add (1.0/fac1, xa);
dxdt *= 1.0 - 1.0/fac1;
dxdt.Add (1.0/fac1, va);
d2xdt2 *= 1.0 - 1.0/fac5;
d2xdt2.Add (1.0/fac5, aa);
t += dt;
}
}
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