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    <p>Here is the source code for a fourth order composition method based on the Verlet scheme. A linear regression of $\log_{10}(\Delta t)$ vs. $\log_{10}(Error)$ will show a slope of 4, as expected from theory (see the graph below). More information can be found <a href="http://www.sciencedirect.com/science/article/pii/0375960190900923" rel="nofollow noreferrer">here</a>, <a href="http://www.cambridge.org/us/academic/subjects/mathematics/computational-science/simulating-hamiltonian-dynamics" rel="nofollow noreferrer">here</a> or <a href="http://www.springer.com/mathematics/computational+science+&amp;+engineering/book/978-3-540-30663-4" rel="nofollow noreferrer">here</a>. </p> <pre><code>#include &lt;cmath&gt; #include &lt;iostream&gt; using namespace std; const double total_time = 5e3; // Parameters for the potential. const double sigma = 1.0; const double sigma6 = pow(sigma, 6.0); const double epsilon = 1.0; const double four_epsilon = 4.0 * epsilon; // Constants used in the composition method. const double alpha = 1.0 / (2.0 - cbrt(2.0)); const double beta = 1.0 - 2.0 * alpha; static double force(double q, double&amp; potential); static void verlet(double dt, double&amp; q, double&amp; p, double&amp; force, double&amp; potential); static void composition_method(double dt, double&amp; q, double&amp; p, double&amp; f, double&amp; potential); int main() { const double q0 = 1.5, p0 = 0.1; double potential; const double f0 = force(q0, potential); const double total_energy_exact = p0 * p0 / 2.0 + potential; for (double dt = 1e-2; dt &lt;= 5e-2; dt *= 1.125) { const long steps = long(total_time / dt); double q = q0, p = p0, f = f0; double total_energy_average = total_energy_exact; for (long step = 1; step &lt;= steps; ++step) { composition_method(dt, q, p, f, potential); const double total_energy = p * p / 2.0 + potential; total_energy_average += total_energy; } total_energy_average /= double(steps); const double err = fabs(total_energy_exact - total_energy_average); cout &lt;&lt; log10(dt) &lt;&lt; "\t" &lt;&lt; log10(err) &lt;&lt; endl; } return 0; } double force(double q, double&amp; potential) { const double r2 = q * q; const double r6 = r2 * r2 * r2; const double factor6 = sigma6 / r6; const double factor12 = factor6 * factor6; potential = four_epsilon * (factor12 - factor6); return -four_epsilon * (6.0 * factor6 - 12.0 * factor12) / r2 * q; } void verlet(double dt, double&amp; q, double&amp; p, double&amp; f, double&amp; potential) { p += dt / 2.0 * f; q += dt * p; f = force(q, potential); p += dt / 2.0 * f; } void composition_method(double dt, double&amp; q, double&amp; p, double&amp; f, double&amp; potential) { verlet(alpha * dt, q, p, f, potential); verlet(beta * dt, q, p, f, potential); verlet(alpha * dt, q, p, f, potential); } </code></pre> <p><img src="https://i.stack.imgur.com/v2cXm.png" alt="Order comparison"></p>
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