/* Simple forward Euler method for velocity integration in time */
void euler(double &x, double &y, double timestep, const FluidQuantity &u, const FluidQuantity &v) const {
double uVel = u.lerp(x, y)/_hx;
double vVel = v.lerp(x, y)/_hx;
x -= uVel*timestep;
y -= vVel*timestep;
}
double maxTimestep() {
double maxVelocity = 0.0;
for (int y = 0; y < _h; y++) {
for (int x = 0; x < _w; x++) {
double u = _u->lerp(x + 0.5, y + 0.5);
double v = _v->lerp(x + 0.5, y + 0.5);
double velocity = sqrt(u*u + v*v);
maxVelocity = max(maxVelocity, velocity);
}
}
double maxTimestep = 4.0*_hx/maxVelocity;
return min(maxTimestep, 1.0);
}
/* Returns the maximum allowed timestep. Note that the actual timestep
* taken should usually be much below this to ensure accurate
* simulation - just never above.
*/
double maxTimestep() {
double maxVelocity = 0.0;
for (int y = 0; y < _h; y++) {
for (int x = 0; x < _w; x++) {
/* Average velocity at grid cell center */
double u = _u->lerp(x + 0.5, y + 0.5);
double v = _v->lerp(x + 0.5, y + 0.5);
double velocity = sqrt(u*u + v*v);
maxVelocity = max(maxVelocity, velocity);
}
}
/* Fluid should not flow more than two grid cells per iteration */
double maxTimestep = 2.0*_hx/maxVelocity;
/* Clamp to sensible maximum value in case of very small velocities */
return min(maxTimestep, 1.0);
}
/* Third order Runge-Kutta for velocity integration in time */
void rungeKutta3(double &x, double &y, double timestep, const FluidQuantity &u, const FluidQuantity &v) const {
double firstU = u.lerp(x, y)/_hx;
double firstV = v.lerp(x, y)/_hx;
double midX = x - 0.5*timestep*firstU;
double midY = y - 0.5*timestep*firstV;
double midU = u.lerp(midX, midY)/_hx;
double midV = v.lerp(midX, midY)/_hx;
double lastX = x - 0.75*timestep*midU;
double lastY = y - 0.75*timestep*midV;
double lastU = u.lerp(lastX, lastY);
double lastV = v.lerp(lastX, lastY);
x -= timestep*((2.0/9.0)*firstU + (3.0/9.0)*midU + (4.0/9.0)*lastU);
y -= timestep*((2.0/9.0)*firstV + (3.0/9.0)*midV + (4.0/9.0)*lastV);
}