source: CIVL/examples/accuracy/diffusion.cvl@ bb03188

/* 1-dimension diffusion equation solver with constant boundaries.
 * Accuracy is first-order in time and second order in space.
 * To verify with CIVL, type:
 *              civl verify diffusion.cvl
 */

$input int n; // number of discrete spatial points
$assume(n >= 1);
$input double dx;  // distance between two consecutive points
$assume(0<dx && dx<1);
$input double dt; // duration of one time step
$assume(0<dt && dt<1);
$input double k;  // diffusivity constant
$assume(k > 0);
$input double u_init[n]; // initial temperatures
// assume u:R^2->R has 4 continuous derivatives in [-1,1]x[0,100]:
$abstract $differentiable(4, [-1,1][0,100]) $real u($real x, $real t);
$input int step; // some arbitrary time step
$assume(step >= 0);
double v[n]; // values at this time step
double v_new[n]; // values at next time step

/* Updates the array v.  The new values of v are computed from the old
 * values of v, and the constants dt, dx, and k.
 */  
void update() {
  // compute the new values in v_new:
  /*@ loop invariant 1<=i && i<=n-1;
    @ loop invariant $forall (int j:1..i-1)
    @   v_new[j] == v[j] + dt*k*(v[j+1] - 2*v[j] + v[j-1])/(dx*dx);
    @ loop assigns i, v_new[1..n-2];
    @*/
  for (int i=1; i<n-1; i++) {
    v_new[i] = v[i] + dt*k*(v[i+1] - 2*v[i] + v[i-1])/(dx*dx);
  }
  // copy v_new back to v:
  /*@ loop invariant 1<=i && i<=n-1;
    @ loop invariant $forall (int j:1..i-1) v[j] == v_new[j];
    @ loop assigns i, v[1..n-2];
    @*/
  for (int i=1; i<n-1; i++) {
    v[i] = v_new[i];
  }     
}

/* This main function is the driver.
 * The annotations assume v[i] is exactly u(i*dx,step*dt) just before
 * updating, and claim that the new v[i] is a very good approximation to
 * u(i*dx,(step+1)*dt).  Specifically, the approximation is
 * O(dt)+O(dx^2) accurate in the interior spatial region. (The endpoints
 * are held constant.)
 */
void main() {
  /*@ loop invariant 0<=i && i<=n;
    @ loop invariant $forall (int j:0..i-1) v[j] == u_init[j];
    @ loop assigns i, v[0..n-1];
    @*/
  for (int i = 0; i < n; i++) {
    v[i] = u_init[i];
  }
  $assume(n*dx < 1);
  $assume(step*dt < 100);
  $assume($forall (int i=0..n-1) v[i] == u(i*dx, step*dt));
  update();
  $assert($uniform (int i=1..n-2) 
    (u(i*dx, (step+1)*dt) - v[i])/dt - $D[u,{t,1}](i*dx, step*dt) + k*$D[u,{x,2}](i*dx, step*dt)
    == $O(dt) + $O(dx*dx));
}