Formulation and numerical computation of the low frequency mean flow of fluids
Final Report Abstract
The oscillatory nature of partial differential equations, which govern the dynamics in the atmosphere and oceans, is a challenge for both understanding the underlying physics and the design of efficient numerical methods. The problems under investigation admit the following form: du/dt + 1/ε Lu + N (u) = 0, where ε is a small parameter, L is a skew-hermitian linear operator, which is responsible for the oscillatory behaviour of the problem, and N is a non-linear term. The fast oscillations have two important consequences. First, the faster the oscillations, the smaller the time step required in the numerical method. Second, the same fast oscillations are also responsible for forming the low frequency mean flow, which is treated in the theory of resonant and near resonant interactions. The subject matter of the project was the low frequency mean flow of fluids, which is a temporal mean flow and represents the behaviour of the fluid on slow temporal scales. For the investigations the rotating shallow water equations (RSWE) and the Boussinesq equations were studied, besides systems of ordinary differential equations which were used as simpler prototype examples, as well as the shallow water equations with bathymetry. The following main outcomes can be stated: 1) New formulations of the mean flow were developed and investigated, using a transformation, related to the method of cancellation of oscillations, and analytical averaging techniques. 2) The novel formulations of the mean flow were used to develop new time integrators of predictor-corrector type for oscillatory differential equations. For the time integrators a multi-grid approach was used. Moreover, the new time stepping methods are parallelisable and therefore promising for computers with large concurrency. An important feature of the new formulations is that they account for the resonant as well as near resonant frequencies due to the non-linearity. Especially, with the new approach it is possible to adjust which modes are slow enough to be part of the mean flow. For the study of the new formulations, investigations on the Boussinseq equations were carried out. The novel mean flow formulations are beneficial for the numerical time stepping schemes, in particular for a good prediction step. The development of the time stepping methods included the following steps: recursive formulation of a multilevel method, which combines the Parareal method, averaging and the transformation; the derivation of an error bound and a model for the computational cost which accounts for the parallelism of the method; implementation of several numerical examples. Encouraging investigations on the efficiency of the multi-level method were carried out, especially for the RSWE.
Publications
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Juliane-Rosemeier/Multi-level-Parareal-Examples: ODE examples solved with the Multi-level Parareal method.
Juliane Rosemeier
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Mean Flow from Phase Averages in the 2D Boussinesq Equations. Atmosphere, 14(10), 1523.
Wingate, Beth A.; Rosemeier, Juliane & Haut, Terry
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Multilevel Parareal Algorithm with Averaging for Oscillatory Problems. SIAM Journal on Scientific Computing, 46(4), A2709-A2736.
Rosemeier, Juliane; Haut, Terry & Wingate, Beth
