Solution of wave propagation problems by the cell discretization method

The Cell Discretization algorithm (CD) is first applied to a fairly complex steady-state wave scattering problem, and the results compared with the same problem solved by the Finite Element Method. (The p.d.e. solved is the elliptic Helmholtz equation for the components of the complex scattered amplitude.) The method is then directly applied to some hyperbolic problems in space-time, using a total discretization (instead of semi-discretization). The problem then becomes one of solving a discretized elliptic equation in successive 'time-slabs' as the process progresses through time. The main difficulty is in choosing the right combination of collocation moments (peculiar to the CD method), intracell polynomials and the 'CFL ratio', i.e., the ratio of the cell length in the time direction to its length(s) in the space direction(s). Apart from this (serious) difficulty, there is little problem in solving several problems involving first and second order wave equations in one and two space dimensions, and one solution of a second order wave equation in two space dimensions. Copious figures are supplied, showing surface plots of the solutions and plots of the spectrum, in the complex plane, of the 'propagation matrix' associated with the problem, are supplied.