void TestConvergence(Integrator const& integrator,
Time const& beginning_of_convergence) {
Length const q_initial = 1 * Metre;
Speed const v_initial = 0 * Metre / Second;
Speed const v_amplitude = 1 * Metre / Second;
AngularFrequency const ω = 1 * Radian / Second;
Instant const t_initial;
#if defined(_DEBUG)
Instant const t_final = t_initial + 10 * Second;
#else
Instant const t_final = t_initial + 100 * Second;
#endif
Time step = beginning_of_convergence;
int const step_sizes = 50;
double const step_reduction = 1.1;
std::vector<double> log_step_sizes;
log_step_sizes.reserve(step_sizes);
std::vector<double> log_q_errors;
log_step_sizes.reserve(step_sizes);
std::vector<double> log_p_errors;
log_step_sizes.reserve(step_sizes);
std::vector<ODE::SystemState> solution;
ODE harmonic_oscillator;
harmonic_oscillator.compute_acceleration =
std::bind(ComputeHarmonicOscillatorAcceleration,
_1, _2, _3, /*evaluations=*/nullptr);
IntegrationProblem<ODE> problem;
problem.equation = harmonic_oscillator;
ODE::SystemState const initial_state = {{q_initial}, {v_initial}, t_initial};
problem.initial_state = &initial_state;
problem.t_final = t_final;
ODE::SystemState final_state;
problem.append_state = [&final_state](ODE::SystemState const& state) {
final_state = state;
};
for (int i = 0; i < step_sizes; ++i, step /= step_reduction) {
integrator.Solve(problem, step);
Time const t = final_state.time.value - t_initial;
Length const& q = final_state.positions[0].value;
Speed const& v = final_state.velocities[0].value;
double const log_q_error = std::log10(
AbsoluteError(q / q_initial, Cos(ω * t)));
double const log_p_error = std::log10(
AbsoluteError(v / v_amplitude, -Sin(ω * t)));
if (log_q_error <= -13 || log_p_error <= -13) {
// If we keep going the effects of finite precision will drown out
// convergence.
break;
}
log_step_sizes.push_back(std::log10(step / Second));
log_q_errors.push_back(log_q_error);
log_p_errors.push_back(log_p_error);
}
double const q_convergence_order = Slope(log_step_sizes, log_q_errors);
double const q_correlation =
PearsonProductMomentCorrelationCoefficient(log_step_sizes, log_q_errors);
LOG(INFO) << "Convergence order in q : " << q_convergence_order;
LOG(INFO) << "Correlation : " << q_correlation;
#if !defined(_DEBUG)
EXPECT_THAT(RelativeError(integrator.order, q_convergence_order),
Lt(0.02));
EXPECT_THAT(q_correlation, AllOf(Gt(0.99), Lt(1.01)));
#endif
double const v_convergence_order = Slope(log_step_sizes, log_p_errors);
double const v_correlation =
PearsonProductMomentCorrelationCoefficient(log_step_sizes, log_p_errors);
LOG(INFO) << "Convergence order in p : " << v_convergence_order;
LOG(INFO) << "Correlation : " << v_correlation;
#if !defined(_DEBUG)
// SPRKs with odd convergence order have a higher convergence order in p.
EXPECT_THAT(
RelativeError(integrator.order + (integrator.order % 2),
v_convergence_order),
Lt(0.02));
EXPECT_THAT(v_correlation, AllOf(Gt(0.99), Lt(1.01)));
#endif
}
void Test1000SecondsAt1Millisecond(
Integrator const& integrator,
Length const& expected_position_error,
Speed const& expected_velocity_error) {
Length const q_initial = 1 * Metre;
Speed const v_initial = 0 * Metre / Second;
Speed const v_amplitude = 1 * Metre / Second;
AngularFrequency const ω = 1 * Radian / Second;
Instant const t_initial;
#if defined(_DEBUG)
Instant const t_final = t_initial + 1 * Second;
#else
Instant const t_final = t_initial + 1000 * Second;
#endif
Time const step = 1 * Milli(Second);
int const steps = static_cast<int>((t_final - t_initial) / step) - 1;
int evaluations = 0;
std::vector<ODE::SystemState> solution;
ODE harmonic_oscillator;
harmonic_oscillator.compute_acceleration =
std::bind(ComputeHarmonicOscillatorAcceleration,
_1, _2, _3, &evaluations);
IntegrationProblem<ODE> problem;
problem.equation = harmonic_oscillator;
ODE::SystemState const initial_state = {{q_initial}, {v_initial}, t_initial};
problem.initial_state = &initial_state;
problem.t_final = t_final;
problem.append_state = [&solution](ODE::SystemState const& state) {
solution.push_back(state);
};
integrator.Solve(problem, step);
EXPECT_EQ(steps, solution.size());
switch (integrator.composition) {
case BA:
case ABA:
EXPECT_EQ(steps * integrator.evaluations, evaluations);
break;
case BAB:
EXPECT_EQ(steps * integrator.evaluations + 1, evaluations);
break;
default:
LOG(FATAL) << "Invalid composition";
}
Length q_error;
Speed v_error;
for (int i = 0; i < steps; ++i) {
Length const q = solution[i].positions[0].value;
Speed const v = solution[i].velocities[0].value;
Time const t = solution[i].time.value - t_initial;
EXPECT_THAT(t, AlmostEquals((i + 1) * step, 0));
// TODO(egg): we may need decent trig functions for this sort of thing.
q_error = std::max(q_error, AbsoluteError(q_initial * Cos(ω * t), q));
v_error = std::max(v_error, AbsoluteError(-v_amplitude * Sin(ω * t), v));
}
#if !defined(_DEBUG)
EXPECT_EQ(expected_position_error, q_error);
EXPECT_EQ(expected_velocity_error, v_error);
#endif
}
