Using an ODE Solver for a Class of Integro-differential Systems

By a simple extension of the Method of Lines, the ordinary differential equation solver VODPK may be used to solve a certain class of integro-differential equation systems (IDE systems). The problems are characterized by a pair of advected frequency-dependent quantities, coupled to a population variable whose rate includes a spectral integral in one space dimension. We have found that with an appropriate choice of preconditioner to aid in the convergence of the linear iterations, an extremely efficient method is obtained for the solution of these types of IDE system problems. We discuss the semidiscretization process and the formation of the preconditioner in some detail. Finally, we present an application of the technique.