# Runge–Kutta time integrators

static double update (scalar * ul, scalar * kl, double t, double dt,
void (* Lu) (scalar * ul, double t, scalar * kl),
scalar * dul, double w)
{
scalar * u1l = list_clone (ul);
foreach() {
scalar u1, u, k;
for (u1,u,k in u1l,ul,kl)
u1[] = u[] + dt*k[];
}
Lu (u1l, t + dt, kl);
foreach() {
scalar du, k;
for (du,k in dul,kl)
du[] += w*k[];
}
delete (u1l), free (u1l);
return w;
}

The runge_kutta() function implements the classical first- (Euler), second- and fourth-order Runge–Kutta time integrators for evolution equations of the form \displaystyle \frac{\partial\mathbf{u}}{\partial t} = L(\mathbf{u}, t) with \mathbf{u} a vector (i.e. list) of evolving fields and L() a generic, user-defined operator.

Given \mathbf{u}, the initial time t, a timestep dt and the function L() which should fill kl with the right-hand-side of the evolution equation, the function below will return \mathbf{u} at time t + dt using the Runge–Kutta scheme specified by order.

void runge_kutta (scalar * ul, double t, double dt,
void (* Lu) (scalar * ul, double t, scalar * kl),
int order)
{
scalar * dul = list_clone (ul);
scalar * kl = list_clone (ul);
Lu (ul, t, kl);
foreach() {
scalar du, k;
for (du,k in dul,kl)
du[] = k[];
}

double w = 1.;
switch (order) {
case 1: // Euler
break;
case 2:
w += update (ul, kl, t, dt, Lu, dul, 1.);
break;
case 4:
w += update (ul, kl, t, dt/2., Lu, dul, 2.);
w += update (ul, kl, t, dt/2., Lu, dul, 2.);
w += update (ul, kl, t, dt,    Lu, dul, 1.);
break;
default:
assert (false); // not implemented
}

foreach() {
scalar u, du;
for (u,du in ul,dul)
u[] += dt/w*du[];
}

delete (dul), free (dul);
delete (kl), free (kl);
}