Stability of Explicit Schemes in the Physical and Frequency Domains

We look at some of the issues involved in designing stable explicit numerical schemes for linear advection equations from two perspectives: (a) in the physical domain, where each scheme represents a particular interpolation of discrete data, and (b) in the frequency domain, where the behavior of each scheme is determined by the spectral characteristics of the operator that is acting on discrete data. We show that (1) the fully discrete form is equivalent to choosing a value for the dependent variable from an interpolation of the data 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, interpolation in the physical domain exceeds the bounds set by the discrete data at a spatial location corresponding to this step size. Comparisons are made between the behavior of the operator in the frequency and physical domains; and the amplification in the frequency domain matches the value of extrema generated by the interpolation. Examples to illustrate both perspectives include first and second difference operators, spatial averaging, and various central and upwind schemes for the linear advection equation.