This program can be extended in the following directions:
<ul>
- <li> Adaptive mesh refinement. As we have seen, we computed everything on a single fixed mesh.
+ <li> Adaptive mesh refinement: As we have seen, we computed everything on a single fixed mesh.
Using adaptive mesh refinement can lead to increased accuracy while not significantly increasing the
computational time.
- <li> Adaptive time-stepping. Although there is no theory about projection methods with variable time-step,
- practice shows that they perform very well. This could be implemented too.
- <li> Variable density incompressible flows. There are projection-like methods for the case of incompressible
- flows with variable density.
+
+ <li> Adaptive time-stepping: Although there apparently is currently no theory about
+ projection methods with variable time step,
+ practice shows that they perform very well.
+
+ <li> Variable density incompressible flows: There are projection-like methods for the case of incompressible
+ flows with variable density. Such flows play a role if fluids of different
+ density mix, for example fresh water and salt water, or alcohol and water.
+
+ <li> Compressible Navier-Stokes equations: These equations are relevant for
+ cases where
+ velocities are high enough so that the fluid becomes compressible, but not
+ fast enough that we get into a regime where viscosity becomes incompressible
+ and the Navier-Stokes equations need to be replaced by the hyperbolic Euler
+ equations of gas dynamics. Compressibility starts to become a factor if the
+ velocity becomes greater than about one third of the speed of sound, so it
+ is not a factor for almost all terrestrial vehicles. On the other hand,
+ commercial jetliners fly at about 85 per cent of the speed of sound, and
+ flow over the wings becomes significantly supersonic, a regime in which the
+ compressible Navier-Stokes equations are not applicable any more
+ either. There are significant applications for the range in between,
+ however, such as for small aircraft or the fast trains in many European and
+ East Asian countries.
</ul>