A Moving Discontinuous Galerkin Finite Element Method for Flows with Interfaces
NAVAL RESEARCH LAB WASHINGTON DC WASHINGTON United States
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A moving discontinuous Galerkin finite element method with interface condition enforcement MDG-ICE 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, while treating the discrete grid geometry as a variable. In contrast to the standard discontinuous Galerkin DG method, this method has both the means to detect, via interface condition enforcement, and 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 non-trivial 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 spacetime 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.
- Fluid Mechanics