A moving discontinuous Galerkin finite element method for flows with interfaces. (26th December 2018)

• Record Type: Journal Article
• Authors: Corrigan, Andrew; Kercher, Andrew D.; Kessler, David A.
• Is Part Of: International journal for numerical methods in fluids. Volume 89:Number 9(2019)
• Page Start: 362
• Page End: 406
• Publication Date: 2018-12-26
• Subjects: discontinuous Galerkin method -- grid adaptivity -- high‐order finite elements -- interface condition enforcement -- shock‐fitting -- space‐time; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4697
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high‐order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space‐time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three‐dimensional flows that provide an initial assessment of the stability and accuracy of the method. Abstract : A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The method preserves a high‐order representation up to the interface, without relying on artificial dissipation or shock capturing for stabilization. The method can compute curved interfaces, with a priori unknown topology, in multiple dimensions. The method applies to both steady flows and unsteady flows via a space‐time formulation.