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Discontinuous Galerkin Methods on Moving Domains with Large Deformations


We present two different numerical approaches for solving compressible flows on moving domains with high-order accuracy. The approaches are base on discontinuous Galerkin (DG) methods and are particularly designed for addressing the large deformation problems as the domain moves.

A moving-mesh technique is first introduced to improve the mesh quality with the domain deforming. The technique moves the mesh nodes by DistMesh algorithm and locally changes the mesh topology by flipping edges or faces, which can be applied in both 2D and 3D. Moreover, some local density control operations are also developed to add or remove the mesh nodes to change the mesh adaptivity.

Our first numerical scheme is formulated on a space-time framework using a nodal DG discretization on space-time domains with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. Along with the numerical scheme, two efficient algorithms for constructing globally conforming space-time slab meshes are given, based on our moving-mesh technique.

The second approach employs DG discreatization with arbitrary-Eulerian-Lagrangian (ALE) framework by solving equations based on smooth mappings. An efficient local L2 projection is used for transferring solutions when mesh topology change happens.

We test our two approaches by a number of numerical cases in both 2D and 3D. The tests involve convergence tests as well as simulations of laminar flows, which shows that the proposed methods achieve high-order accuracy and are able to handle problems with complex geometric motions.

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