src/test/kuramoto.c

    Kuramoto–Sivashinsky equation

    Duchemin et Eggers, 2014, section 6 propose to use their “Explicit-Implicit-Null” method to solve the Kuramoto–Sivashinsky equation \displaystyle \partial_t u = -u \partial_x u - \partial^2_x u - \partial^4_x u while avoiding the stringent explicit timestep restriction due to the fourth derivative.

    In this example we show that this can also be done using the implicit multigrid solver.

    #include "grid/multigrid1D.h"
#include "solve.h"

    This is the simple explicit discretisation (which is not used).

    void solve_explicit (scalar u, double dt)
{
  scalar du[];
  foreach()
    du[] = - u[]*(u[1] - u[-1])/(2.*Delta)
      - (u[-1] - 2.*u[] + u[1])/sq(Delta)
      - (u[-2] - 4.*u[-1] + 6.*u[] - 4.*u[1] + u[2])/sq(sq(Delta));
  foreach()
    u[] += dt*du[];
}

int main()
{

    We reproduce the same test case as in section 6.2 of Duchemin & Eggers.

      init_grid (512);
  L0 = 32.*pi;
  periodic (right);
  scalar u[];
  foreach()
    u[] = cos(x/16.)*(1. + sin(x/16.));

    The timestep is set to 0.1, which is significantly larger than that in Duchemin & Eggers (0.014).

      double dt = 1e-1;
  //  double dt = 1.4e-4;
  int i = 0;
  TOLERANCE = 1e-6;
  for (double t = 0; t <= 150; t += dt, i++) {
    if (i % 1 == 0) {
      foreach()
	fprintf (stdout, "%g %g %g\n", t, x, u[]);
      fputs ("\n", stdout);
    }
    scalar b[];
    foreach()
      b[] = u[] - dt*u[]*(u[1] - u[-1])/(2.*Delta);
    solve (u,
	   u[] + dt*(u[-1] - 2.*u[] + u[1])/sq(Delta)
	   + dt*(u[-2] - 4.*u[-1] + 6.*u[] - 4.*u[1] + u[2])/sq(sq(Delta)),
	   b);
    fprintf (stderr, "%g %d\n", t, solve_stats.i);
    //    solve_explicit (u, dt);
  }
}

    The result can be compared to Figure 8 of Duchemin & Eggers.

    set term PNG enhanced font ",10"
set output 'sol.png'
set pm3d map
set xlabel 'x'
set ylabel 't'
set xrange [100:0]
set yrange [0:150]
splot 'out' u 2:1:3
    Solution of the Kuramoto–Sivashinski equation (script)

    Solution of the Kuramoto–Sivashinski equation (script)

    References

    [duchemin2014]

    Laurent Duchemin and Jens Eggers. The explicit–implicit–null method: Removing the numerical instability of pdes. Journal of Computational Physics, 263:37–52, 2014. [ .pdf ]