Numerical simulation of differential systems displaying rapidly oscillating solutions
This paper is concerned with the numerical approximation of some systems of differential equations whose solutions contain components with very different time scales. The systems under consideration have the form \( \dot u + A u + \varepsilon^{-1} L u + B(u,u) = f \) where \( A\) is a symmetric positive definite matrix, \(L\) is either a symmetric positive definite or antisymmetric matrix that can be diagonalized, \(B\) is a quadratic operator and \( \varepsilon >0\) is a small parameter. To approximate the solutions of these systems the authors introduce a previous averaging process that leads to a new system whose solutions are slowly varying and therefore can be integrated numerically by a standard numerical integrator without severe restrictions on the size of the step. Then the solution of the original system can be expressed in terms of the solution of the averaged system. Some theoretical results are stated to justify the proposed approach. Finally some low-dimensional examples are presented to show the behaviour of the new approach. From these examples it follows that, for small \(\varepsilon\), the approximate solutions obtained are able to capture the main features of the solution with low computational effort.
scientific article; zbMATH DE number 1282085