sandbox/zaleski/centralscheme.h

Explicit central scheme for the 1D inviscid Burgers equation

Solves \partial_t u + \partial_x \left(\frac{u^2}{2}\right) = 0 using a first-order forward Euler time step with a centered spatial difference for the flux divergence: u^{n+1}_i = u^n_i - \frac{\tau}{2\Delta} \left[ F(u^n_{i+1}) - F(u^n_{i-1}) \right] where F(u) = u^2/2.

#include "run.h"
#include "timestep.h"

Field declaration

scalar u[];

Flux function

The Burgers flux F(u) = u^2/2.

#define F_burgers(u) (sq(u)/2.)

Default parameters

event defaults (i = 0)
{
  CFL = 0.4;
}

Stability and timestep

The CFL condition for Burgers is \tau \leq \Delta / \max|u|.

double dtmax;

event init (i = 0)
{
  dtmax = DT;
  event ("stability");
}

event set_dtmax (i++, last) dtmax = DT;

event stability (i++, last)
{
  vector vel[];
  foreach()
    vel.x[] = u[];
  dt = dtnext (timestep (vel, dtmax));
}

Time integration

Forward Euler with centered flux difference.

event time_step (i++, last)
{
  scalar un[];

Save the current field before updating.

  foreach()
    un[] = u[];

Update with centered difference of the flux: u^{n+1}_i = u^n_i - \frac{\tau}{2\Delta}\left[F(u^n_{i+1}) - F(u^n_{i-1})\right]

  foreach()
    u[] = un[] - (dt/(2.*Delta)) * (F_burgers(un[1]) - F_burgers(un[-1]));
}