Discontinuous Galerkin finite element in time for solving periodic differential equations

The dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing. Discontinuous Galerkin (DG) methods which are known to be higher order accurate, A-stable and locally conservative are used to solve such equations in this work. DG methods are shown to find the steady state periodic solution of (a) constant coefficient structural dynamics problem with periodic forcing and (b) helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing. Convergence of the numerical solution obtained is studied, for both the above equations, while h and p are refined separately. It is found that the DG method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes. It is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p > 12. Two adaptive strategies based on the element level displacement jump norm, i.e., h and hp refinement, are proposed and demonstrated on problems with higher harmonic forcing, primary controls and parametric variations.