Ascent

A fast and flexible C++ simulation engine and differential equation solver.

Link	GitHub repository
Language	C++
License	Apache-2.0 license

Lorenz equations

We would like to solve the Lorenz equations, a classical chaotic system given by:

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

with parameter \(\sigma=10\), \(\rho = 28\) and \(\beta = \frac{8}{3}\), and the initial state \((x_0, y_0, z_0) = (1,1,1)\). We solve it with a classical Runge-Kutta method of order 4, given by asc::RK4 method.

To solve a problem with Ascent we first write our equation as a ODE of the form:

\[ \dot{u} = f(t, u) \]

and the user provide the function \(f\) as (a lambda function for the example):

  auto f = [](state_t const& u, state_t & du, double t);

where u the current state of the function, t the current time and output du \(f(t,u)\) by reference. By default, Ascent provide a asc::state_t type as a reference to std::vector<double> class.

    auto lorenz = [&]( asc::state_t const& y, asc::state_t& dy, double t )
    {
        dy[0] = sigma * ( y[1] - y[0] );
        dy[1] = y[0] * ( rho - y[2] ) - y[1];
        dy[2] = y[0] * y[1] - beta * y[2];
    };

If we need a more generic state type we need to use asc::RK4T<state_t> integrator in place of asc::RK4. Next we can call this integrator for one step with the time step dt by calling it as

  integrator(f, un, tn, dt);

which takes current state un and time tn by references (also dt if this is an adaptive time step method), inside the time loop:

    while ( tn < tf )
    {
        vec_observer( yn, tn );
        integrator( lorenz, yn, tn, dt );
    }

where vec_observer is an observer to save states in a std::vector.

For the complet example, see lorenz.cpp source file.

Transport equation

In this example we would like to solve the following PDE:

\[ \partial_t u + a \partial_x u = 0 \]

with \(t>0\), on the torus \(x\in[0, 1)\), a velocity \(a=1\) and the initial condition given by a hat function:

\[\begin{split} u(0, x) = \begin{cases} x - 0.25 & \text{if } x\in[0.25, 0.5[ \\ -x + 0.75 & \text{if } x\in[0.5, 0.75[ \\ 0 & \text{else} \end{cases} \end{split}\]

We choose a first order up-wind scheme to estimate the \(x\) derivative and a forward Euler method for the time discretization given by asc::Euler method.

We define the up-wind scheme as:

    auto upwind = [a, n_x, dx]( asc::state_t const& y, asc::state_t& dy, double t )
    {
        dy[0] = -( std::max( a, 0. ) * ( y[0] - y[n_x - 1] ) + std::min( a, 0. ) * ( y[1] - y[0] ) ) / dx;

        for ( std::size_t i = 1; i < n_x - 1; ++i )
        {
            dy[i] = -( std::max( a, 0. ) * ( y[i] - y[i - 1] ) + std::min( a, 0. ) * ( y[i + 1] - y[i] ) ) / dx;
        }

        dy[n_x - 1] = -( std::max( a, 0. ) * ( y[n_x - 1] - y[n_x - 2] ) + std::min( a, 0. ) * ( y[0] - y[n_x - 1] ) ) / dx;
    };

The time loop is the same as for Lorenz equation.

For the complet example, see transport.cpp source file.

Arenstorf orbit

The Arenstorf orbit problem is a classical problem to test adaptive time step methods:

\[\begin{split} \begin{cases} \ddot{x} &= x + 2\dot{y} - \frac{1-\mu}{r_1^3}(x+\mu) - \frac{\mu}{r_2^3}(x-1+\mu) \\ \ddot{y} &= y - 2\dot{x} - \frac{}{1-\mu}{r_1^3}y - \frac{\mu}{r_2^3}y \end{cases} \end{split}\]

with initial condition \((x,\dot{x},y,\dot{y})=(0.994, 0, 0, -2.001585106)\), \(r_1\) and \(r_2\) given by

\[ r_1 = \sqrt{(x+\mu)^2 + y^2},\quad r_2 = \sqrt{(x-1+\mu)^2 + y^2} \]

and with parameter \(\mu = 0.012277471\).

First of all, we need to rewrite this problem into a first order derivative equation in time

\[\begin{split} \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \end{pmatrix} = \begin{pmatrix} x \\ y \\ \dot{x} \\ \dot{y} \end{pmatrix}, \qquad \begin{cases} \dot{y}_1 = y_3 \\ \dot{y}_2 = y_4 \\ \dot{y}_3 = y_1 + 2y_4 - \frac{1-\mu}{r_1^3}(y_1 + \mu) - \frac{\mu}{r_2^3}(y_1-1+\mu) \\ \dot{y}_4 = y_2 - 2y_3 - \frac{1-\mu}{r_1^3}y_2 - \frac{\mu}{r_2^3}y_2 \\ \end{cases} \end{split}\]

We define this system as:

    auto arenstorf = [=]( asc::state_t const& y, asc::state_t& dy, double t )
    {
        double const y1 = y[0];
        double const y2 = y[1];
        double const y3 = y[2];
        double const y4 = y[3];

        double const r1 = sqrt( ( y1 + mu ) * ( y1 + mu ) + y2 * y2 );
        double const r2 = sqrt( ( y1 - 1. + mu ) * ( y1 - 1. + mu ) + y2 * y2 );

        dy[0] = y3;
        dy[1] = y4;
        dy[2] = y1 + 2. * y4 - ( 1. - mu ) * ( y1 + mu ) / ( r1 * r1 * r1 ) - mu * ( y1 - 1. + mu ) / ( r2 * r2 * r2 );
        dy[3] = y2 - 2. * y3 - ( 1. - mu ) * y2 / ( r1 * r1 * r1 ) - mu * y2 / ( r2 * r2 * r2 );
    };

We solve this example with given method asc::DOPRI45 which is the method RK5(4) 7M in [DP80] (mainly call DOPRI5). For this we need to set the integrator and settings for adaptive time step method

    asc::DOPRI45 integrator;
    asc::AdaptiveT<double> adaptive_settings( 1e-5, 1e-5, 0.9 );

then the time loop becomes

    while ( tn < tf )
    {
        vec_observer( yn, tn, dt );
        integrator( arenstorf, yn, tn, dt, adaptive_settings );
    }

where we should pass the adaptive_setting object to integrator.

For the complet example, see arenstorf.cpp source file.