Eulerian and Lagrangian Methods

In the methods discussed so far the differential equations have been discretised by defining the value of the unknown at fixed points in space. Such methods are known as Eulerian methods. We have confined ourselves to a regular grid of points, but sometimes it is advantageous to choose some other grid. An obvious example is when there is some symmetry in the problem, such as cylindrical or spherical: often it is appropriate to base our choice of grid on cylindrical or spherical coordinates rather than the Cartesian ones used so far. Suppose, however, we are dealing with a problem in electromagnetism or optics, in which the dielectric constant varies in space. Then it might be appropriate to choose a grid in which the points are more closely spaced in the regions of higher dielectric constant. In that way we could take account of the fact that the wavelengths expected in the solution will be shorter. Such an approach is known as an adaptive grid. In fact it is not necessary for the grid to be fixed in space. In fluid mechanics, for example, it is often better to define a volume of space containing a fixed mass of fluid and to let the boundaries of these cells move in response to the dynamics of the fluid. The differential equation is transformed into a form in which the variables are the positions of the boundaries of the cells rather than the quantity of fluid in each cell. A simple example of such a Lagrangian method is described in the Lagrangian Fluid project.