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Andrew Winters: Modern discontinuous Galerkin methods for computational fluid dynamics

Time: Thu 2022-03-24 14.00 - 15.00

Location: KTH, Seminar room 3721, Lindstedsvägen 25

Participating: Andrew Winters (Linköping)

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Abstract: Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of hyperbolic partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Over the past decade, DG has undergone an extensive transformation into its modern form; capable of robust, adaptive simulations for complex transient flows.

Solutions of non-linear conservation laws contain many complex phenomena such as discontinuities, singularities, and turbulence. These phenomena are all time dependent and feature multiple scales in space and time. A wide variety of low-order and high-order numerical methods have been developed over many decades. There is, however, a balancing act when applying either type of method to a given problem:

1) Low-order methods offer remarkable robustness but require a very large number of degrees of freedom (DOFs) to properly capture multi-scale non-linear phenomena.

2) High-order methods offer great capabilities to accurately capture non-linear phenomena while requiring a moderate number of DOFs. However, they often lack robustness.

One approach for a shock capturing framework is to blend a low-order finite volume method with a high-order nodal DG method. Through carful construction of the geometric terms, this strategy is generalisable to multi-dimensional curvilinear meshes.

The aim of this talk is to dissect and discuss a modern form of nodal DG spectral element methods. These DG methods combine favourable features of other methods, e.g., geometric flexibility of finite element methods, skew-symmetric formulations of finite difference methods, and entropy stable numerical fluxes from finite volume methods. Implementation aspects of this modern DG formulation will also be included.