High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Shu, Chi‐Wang

doi:10.1016/j.jcp.2016.04.030

Cited by 161 publications

(86 citation statements)

References 286 publications

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“…In the present study we employ fifth-order WENO scheme (WENO5) introduced by Coralic and Colonius [10], unless otherwise noted. High-order WENO schemes are, in general, robust in capturing discontinuities including shock wave and material's interface, while capable of resolving continuous waves with a high-amplitude with relatively small numerical dissipation/dispersion [48,49,50]. Such properties of WENO schemes are suitable for simulations of cloud cavitation in the regime of the interest of the present study; a passage of strong pressure waves causes violent, nonlinear oscillations of bubbles, each of which emits strong pressure waves with broadband frequency and generate complex structures of pressure fields by mutual interactions.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…In the cases with β 0 = 5.0 × 10 −6 and 1.0 × 10 −5 , C T Emp /C T w fluctuates rapidly and grows to a value of 10, while in the cases with higher values of β 0 , the fluctuation is smaller and the value stays close to 1. Fig 8b shows the averaged value of C T Emp /C T w within the interval of t = [20,50] µs as a function of β 0 and N C , where N C is the averaged number of bubbles contained in the region that the operator S averages over (see equation (52)). In accordance with fig 8a, with β 0 = 5.0 × 10 −6 and 1.0 × 10 −5 , C T Emp /C T w takes a value much larger than 1, while with β 0 higher than 2.0 × 10 −5 it takes a value close to 1.…”

Section: Bubble Screen Problemmentioning

confidence: 99%

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Eulerian–Lagrangian method for simulation of cloud cavitation

Maeda

Colonius

2018

Journal of Computational Physics

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We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is tracked by solving the Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid as the void fraction by using a regularization (smearing) kernel. In the most general case, where the bubble distribution is arbitrary, three-dimensional Cartesian grids are used for spatial discretization. In order to reduce the computational cost for problems possessing translational or rotational homogeneities, we spatially average the governing equations along the direction of symmetry and discretize the continuous phase on two-dimensional or axi-symmetric grids, respectively. We specify a regularization kernel that maps the three-dimensional distribution of bubbles onto the field of an averaged two-dimensional or axi-symmetric void fraction. A closure is developed to model the pressure fluctuations at the sub-grid scale as synthetic noise. For the examples considered here, modeling the sub-grid pressure fluctuations as white noise agrees a priori with computed distributions from three-dimensional simulations, and suffices, a posteriori, to accurately reproduce the statistics of the bubble dynamics. The numerical method and its verification are described by considering test cases of the dynamics of a single bubble and cloud cavitaiton induced by ultrasound fields. (Kazuki Maeda) arXiv:1712.00670v2 [physics.flu-dyn] 5 Dec 2017 such methods are the most suitable for simulations at the scales of bubbles for a short period of time, typically single events of the collapse of bubbles [17,18,19,20].For more complex bubbly mixtures with relatively low void fractions of O(10 −2 ), methods that solve volume-or ensemble-averaged equations of motion have been used to compute the propagation of acoustic and shock waves [21,22,23,24,25,26,27,28,29,30] and the dynamics of bubble clouds [31,32,33,34,35,36,37]. In the classical averaging approaches, the bubbly-mixture is treated as a continuous media. The volume of dispersed bubbles is converted to a continuous void fraction field in a control volume that contains a sufficiently large number of bubbles. The length-scale of the control volume (averaging length-scale) is larger than the characteristic inter-bubble distance. The dynamics of the gas-phase are closed by considering the averaged change in the volumetric oscillations of bubbles in response to pressure fluctuations in the mixture. Bubbles are typically modeled as spherical cavities, of which dynamics are described by ordinary...

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Section: Spatial Discretizationmentioning

confidence: 99%

Section: Bubble Screen Problemmentioning

confidence: 99%

Eulerian–Lagrangian method for simulation of cloud cavitation

Maeda

Colonius

2018

Journal of Computational Physics

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“…Importantly, DG methods exhibit favorable properties when collisions with a background are included, as they recover the correct asymptotic behavior in the diffusion limit, characterized by frequent collisions (e.g., [34,35,36]). The DG method was introduced in the 1970s by Reed & Hill [37] to solve the neutron transport equation, and has undergone remarkable developments since then (see, e.g., [38] and references therein).We are concerned with the development and application of DG methods for the fermionic two-moment model that can preserve the aforementioned algebraic constraints and ensure realizable moments, provided the initial condition is realizable. Our approach is based on the constraint-preserving (CP) framework introduced in [1], and later extended to the Euler equations of gas dynamics in [2].…”

mentioning

confidence: 99%

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Chu

Endeve

Hauck

et al. 2019

Journal of Computational Physics

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Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f . The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky & Bludman [3] and Banach & Larecki [4]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., f ∈ [0, 1]). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and -without interactions with the background -reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher [5]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport. (Ran Chu), endevee@ornl.gov (Eirik Endeve), hauckc@ornl.gov (Cory D. Hauck), mezz@utk.edu (Anthony Mezzacappa) both spectral and finite volume methods and are an attractive option for solving hyperbolic partial differential equations (PDEs). They achieve high-order accuracy on a compact stencil; i.e., data is only communicated with nearest neighbors, regardless of the formal order of accuracy, which can lead to a high computation to communication ratio, and favorable parallel scalability on heterogeneous architectures has been demonstrated [32]. Furthermore, they can easily be applied to problems involving curvilinear coordinates (e.g., beneficial in numerical relativity [33]). Importantly, DG methods exhibit favorable properties when collisions with a background are included, as they recover the correct asymptotic behavior in the diffusion limit, characterized by frequent collisions (e.g., [34,35,36]). The DG method was introduced in the 1970s by Reed & Hill [37] to solve the neutron transport equation, and has undergone remarkable developments since then (see, e.g., [38] and references therein).We are concerned with t...

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“…For more details about these subjects, we refer to previous studies. [32][33][34][35][36] To begin with, we apply a 1-dimensional scalar hyperbolic conservation law to present the DG method…”

Section: A Review Of High Order Dg Methods and Finite Volume Weno Schemementioning

confidence: 99%

Well‐balanced discontinuous Galerkin method and finite volume WENO scheme based on hydrostatic reconstruction for blood flow model in arteries

Delestre

2017

Numerical Methods in Fluids

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Summary The blood flow model in arteries admits the steady state solutions, for which the flux gradient is nonzero, and is exactly balanced by the source term. In this paper, by means of hydrostatic reconstruction, we construct a high order discontinuous Galerkin method, which exactly preserves the dead‐man steady state, which is characterized by a discharge equal to zero (analogue to hydrostatic equilibrium). Moreover, the method maintains genuine high order of accuracy. Subsequently, we apply the key idea to finite volume weighted essentially non‐oscillatory schemes and obtain a well‐balanced finite volume weighted essentially non‐oscillatory scheme. Extensive numerical experiments are performed to verify the well‐balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

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High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Cited by 161 publications

References 286 publications

Eulerian–Lagrangian method for simulation of cloud cavitation

Eulerian–Lagrangian method for simulation of cloud cavitation

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Well‐balanced discontinuous Galerkin method and finite volume WENO scheme based on hydrostatic reconstruction for blood flow model in arteries

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