where U(t), an N vector is the spatial discretization of u, and A is a N Ο« N matrix whose nonzero entries form a
We look at some of the issues involved in designing stable explicit numerical schemes for linear advection equations from two per-narrow band around the diagonal. Boundary conditions spectives: (a) in the physical domain, where each scheme represents specified at x Ο 0 and x Ο L also affect the eigenvalue a particular interpolation of discrete data, and (b) in the frequency spectrum of A.
domain, where the behavior of each scheme is determined by the
The fully discrete form of the equation is obtained on spectral characteristics of the operator that is acting on discrete data.
discretizing the temporal behavior. At time level n Ο© 1, U
We show that (1) the fully discrete form is equivalent to choosing a value for the dependent variable from an interpolation of the data is expressed as in the spatial domain at the previous time level, (2) interpolation generates a continuous function (polynomial) in the physical space,
(3) size of the time step used in updating the solution determines the location from where the interpolated value is obtained, and (4) if a choice of step size shows amplification in the spectral domain, where U n j is the jth component of U at time level n.