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Multilevel time integrators for large-scale atmospheric flows

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Multiscale Numerics for the Atmosphere and Ocean

The fluid dynamically most comprehensive mathematical model of the atmosphere, the Euler equations, admits solutions driven by compressibility, buoyancy, and inertia. In the low Mach number case there is a scale separation of the three effects, and reduced sets of equations have been developed from the fully compressible model to describe the different regimes. In a context of increasing computing resources, reliable discrete solvers have to be devised, which can resolve the different scales and conserve of physical quantities as mass and total energy. As for the time discretization, one faces the choice between the stability-constrained explicit methods and the unconditionally stable, but costly and overdispersive implicit methods. Semi- implicit methods aim at exploiting the advantages of the two approaches as well as reducing their weak points. The goal of the work we present is to obtain a second-order accurate method to reproduce multiscale features of solutions of the fully compressible equations, filtering unwanted small-scale disturbances but retaining the properties of significant waves. An anelastic code for small-scale flows discretised with a projection method is extended with the insertion of an implicit pressure term; as a result, an additional zero-order term is added to the Poisson equation for the pressure in the correction step. On one hand, the approach is in agreement with a standard discretisation of the wave equation for the pressure; on the other hand, the discretisation reduces to the anelastic one for vanishing Mach number. Preliminary runs on advection test cases confirm the feasibility of the approach; large-scale tests will be performed with the insertion of the Coriolis term and the adoption of a suitable spherical grid. Then, a multilevel time discretisation based on multigrid techniques will enable to simulate multiscale test cases, thereby paving the way for an accurate and efficient all-speed solver.

This talk is part of the Isaac Newton Institute Seminar Series series.

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