Lorenz system

\[\begin{split}\begin{cases} \dot{x} = \sigma(y-x) \\ \dot{y} = \rho x - y - xz \\ \dot{z} = xy - \beta z \\ \end{cases}\end{split}\]

with initial condition \((x,y,z)=(1,1,1)\) and parameter \(\sigma=10\), \(\rho=28\) and \(\beta=\frac{8}{3}\).

Runge-Kutta solver

First we need to write a problem:

using state_t = std::valarray<double>;

state_t lorenz(double t, const state_t& u)
{
  double sigma=10. , rho=28., beta=8./3.;
  return {
    signa*( u[1] - u[0] ),
    rho*u[0] - u[1] - u[0]*u[2],
    u[0]*u[1] - beta*u[2]
  };
}

//...

auto pb_rk = ode::make_problem(lorenz);

Note

We can also use a lambda to define a problem, or a collection of function as we present in the following part about splitting methods

Next we define the initial condition and call ode::solve():

state_t u0{1.,1.,1.};

using namespace observer; // to use _fobs litteral

ode::solve( pb_rk , ode::runge_kutta::rk33() , u0 , {0.,20.} , 0.01 , "lorenz_rk33.dat"_fobs );