Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

Santis, Dante De; Geraci, Gianluca; Guardone, Alberto

doi:10.1002/fld.3862

Cited by 3 publications

(1 citation statement)

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“…Johnsen & Colonius [10,11] used cylindrical coordinates with azimuthal symmetry to simulate the collapse of an initially spherical gas bubble in shock-wave lithotripsy by solving the Euler equations inside and outside the bubble using WENO. De Santis [5] showed equivalence between their Lagrangian finite element and finite volume schemes in cylindrical coordinates. Xing and Shu [24][25][26] performed extensive studies of hyperbolic systems with source terms, which are relevant as the equations in cylindrical/spherical coordinates can be written with geometrical source terms.…”

Section: Introductionmentioning

confidence: 99%

High order schemes for cylindrical/spherical coordinates with radial symmetry

Wang

Johnsen

2013

21st AIAA Computational Fluid Dynamics Conference

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We consider implementations of high-order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for the Euler equations in cylindrical and spherical coordinate systems with radial dependence only. The main concern of this work lies in ensuring both high-order accuracy and conservation. Three different spatial discretizations are assessed: one that is shown to be high-order accurate but not conservative, one conservative but not high-order accurate, and a new approach that is both high-order accurate and conservative. For cylindrical and spherical coordinates, we present convergence results for the advection equation and the Euler equations with an acoustics problem; we then use the Sod shock tube and the Sedov point-blast problems in cylindrical coordinates to verify our analysis and implementations.Akhatov, et al.[1] used a first-order Godunov scheme to simulate liquid flow outside of a single bubble whose radius was given by the Rayleigh-Plesset equation. This approach assumes spherical symmetry but does not solve the equations of motion inside the bubble.High-order accurate methods are becoming mainstream in computational fluid dynamics [23]. However, implementation of such methods in cylindrical/spherical geometries is not trivial. Several recent studies in cylindrical and spherical coordinates have focused on the Lagrangian form of the equations [14,21]. The Euler equations in cylindrical or spherical geometry were studied by Maire [14] using a cell-centered Lagrangian scheme, which ensures conservation of momentum and energy. These equations were also considered by Omang et al. [16] using Smoothed Particle Hydrodynamics (SPH), though SPH methods are generally not high-order accurate. On the other hand, solving the equations in Eulerian form is not trivial, especially when trying to ensure conservation and high-order accuracy. Li [12] attempted to implement Eulerian finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes [8] on cylindrical and spherical grids, but did not achieve satisfactory results in terms of accuracy and conservation. Liu et al.[7] considered flux difference-splitting methods for ducts with area variation. [13] followed a similar formulation of the equations employing a total variation diminishing method to simulate explosions in air. Johnsen & Colonius [10,11] used cylindrical coordinates with azimuthal symmetry to simulate the collapse of an initially spherical gas bubble in shock-wave lithotripsy by solving the Euler equations inside and outside the bubble using WENO. De Santis [5] showed equivalence between their Lagrangian finite element and finite volume schemes in cylindrical coordinates. performed extensive studies of hyperbolic systems with source terms, which are relevant as the equations in cylindrical/spherical coordinates can be written with geometrical source terms. Although one of their test cases involved radial flow in a nozzle using the quasi one-dimensional nozzle flow equations, they did not consider the general gas dyn...

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