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Maljaars, Jakob M.; Labeur, Robert Jan; Trask, Nathaniel; Sulsky, Deborah

Conservative, high-order particle-mesh scheme with applications to advection-dominated flows. (English) Zbl 1440.76079

Comput. Methods Appl. Mech. Eng. 348, 443-465 (2019).
Summary: By combining concepts from particle-in-cell (PIC) and hybridized discontinuous Galerkin (HDG) methods, we present a particle-mesh scheme for flow and transport problems which allows for diffusion-free advection while satisfying mass and momentum conservation – locally and globally – and extending to high-order spatial accuracy. This is achieved via the introduction of a novel particle-mesh projection operator which casts the particle-mesh data transfer as a PDE-constrained optimization problem, permitting advective flux functionals at cell boundaries to be inferred from particle trajectories. This optimization problem seamlessly fits in a HDG framework, whereby the control variables in the optimization problem serve as advective fluxes in the HDG scheme. The resulting algebraic problem can be solved efficiently using static condensation. The performance of the scheme is demonstrated by means of numerical examples for the linear advection-diffusion equation and the incompressible Navier-Stokes equations. The results demonstrate optimal spatial accuracy, and when combined with a \(\theta\) time integration scheme, second-order temporal accuracy is shown.

Cited in 1 Document

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids

Keywords:

hybridized discontinuous Galerkin; finite element methods; particle-in-cell; PDE-constrained optimization; conservation; advection-dominated flows

Software:

FEniCS
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Full Text: DOI

References:

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