A numerical method for solving systems of linear ordinary differential equations with rapidly oscillating solutions (Q1184621)

The authors present a numerical method which allows the accurate and efficient solution of systems (*) \(dz_ i/dx=\sum^ N_{j=1}A_{ij}(x)z_ j(x)\), \(i=1(1)N\), of linear ordinary differential equations when the solutions vary rapidly relative to the coefficients of the variables. For a typical solution which oscillates with many nodes and with rapidly varying wavelength and amplitudes, this method has many advantages over the usual solution by direct or approximation techniques. For systems (*) of higher order the usual method of solution is direct numerical integration of the problem using a standard package (e.g. an adaptive Runge-Kutta technique). But this approach has a number of drawbacks. If high accuracy is required then about 50 points per local characteristic length of the solution must be taken to achieve about 4- figure accuracy and the computational overhead goes up roughly linearly with the number of oscillations. Moreover, to generate many solutions the task of storing the results or computing the required functions (while integrating) is usually prohibitive. The method presented by the authors does not suffer from these disadvantages. It involves a computational overhead which is substantially independent of the local wavelength and presents the solution in terms of quantities which vary on the intrinsic scalelength of the coefficient matrices \(A_{ij}(x)\) and hence can be much more easily stored for other purposes. The method is shown to work quite well. It is applied to the problem of calculating eigenfrequencies and eigenmodes for the nonradial oscillations in stars. The results of the authors' method for this fourth-order system eigenvalue problem are compared with those obtained by a shooting technique employing a Runge-Kutta integrator. The process involves the numerical development of a linearly independent set of solutions of the governing system of differential equations which span the solution space. They have the form of asymptotic solutions appropriate to the short local scalelength limit plus remainder. For details the reader is referred to the paper.