Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments

Shu, Chi‐Wang

doi:10.1007/978-3-319-01818-8_2

Cited by 44 publications

(19 citation statements)

References 106 publications

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“…The DG finiteelement method was originally proposed byReed and Hill in 1973 [31]. A more recent survey of DG methods is given by Shu[32]. The DG method developed in this thesis…”

mentioning

confidence: 99%

A Non-parametric Discontinuous Galerkin Formulation of the Integral Boundary Layer Equations with Strong Viscous/Inviscid Coupling

Zhang

Galbraith

Allmaras

et al. 2017

23rd AIAA Computational Fluid Dynamics Conference

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A non-parametric discontinuous Galerkin (DG) finite-element formulation is developed for the integral boundary layer (IBL) equations with strong viscous-inviscid coupling. This DG formulation eliminates the need of explicit curvilinear coordinates in traditional boundary layer solvers, and thus enables application to complex geometries even involving non-smooth features. The usual curvilinear coordinates are replaced by a local Cartesian basis, which is conveniently constructed in the DG finite-element formulation. This formulation is also applicable to the general convection-source type of partial differential equations defined on curved manifolds. Other benefits of DG methods are maintained, including support for high-order solutions and applicability to general unstructured meshes. For robust solution of the coupled IBL equations, a strong viscous-inviscid coupling scheme is also proposed, utilizing a global Newton method. This method provides for flexible and convenient coupling of viscous and inviscid solutions, and is readily extensible to coupling with more disciplines, such as structural analysis. As a precursor to the three-dimensional strongly-coupled IBL method, a two-dimensional IBL solver coupled with a panel method is implemented. Numerical examples are presented to demonstrate the viability and utility of the proposed methodology.

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“…The DG finiteelement method was originally proposed byReed and Hill in 1973 [31]. A more recent survey of DG methods is given by Shu[32]. The DG method developed in this thesis…”

mentioning

confidence: 99%

A Non-parametric Discontinuous Galerkin Formulation of the Integral Boundary Layer Equations with Strong Viscous/Inviscid Coupling

Zhang

Galbraith

Allmaras

et al. 2017

23rd AIAA Computational Fluid Dynamics Conference

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“…Jyoti and Srivastava deployed a four‐step wavelet Galerkin Method for parabolic and hyperbolic problems. Chi used the discontinuous Galerkin method for time‐dependent partial differential equations in medical transport problems.…”

Section: Finite Element Simulation With Freefem++mentioning

confidence: 99%

Finite element analysis of non‐Newtonian magnetohemodynamic flow conveying nanoparticles through a stenosed coronary artery

Vasu

Dubey

Bég

2019

Heat Trans. Asian Res.

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The present study considers two-dimensional mathematical modeling of non-Newtonian nanofluid hemodynamics with heat and mass transfer in a stenosed coronary artery in the presence of a radial magnetic field. The second-grade differential viscoelastic constitutive model is adopted for blood to mimic non-Newtonian characteristics, and blood is considered to contain a homogenous suspension of nanoparticles. The Vogel model is employed to simulate the variation of blood viscosity as a function of temperature.The governing equations are an extension of the Navier-Stokes equations with linear Boussinesq's approximation and Buongiorno's nanoscale model (which simulates both heat and mass transfer). The conservation equations are normalized by employing appropriate nondimensional variables. It is assumed that the maximum height of the stenosis is small in comparison with the radius of the artery, and, furthermore, that the radius of the artery and length of the stenotic region are of comparable magnitude. To study the influence of vessel geometry on blood flow and nanoparticle transport, variation in the design and size of the stenosis is considered in the domain. The transformed equations are solved numerically by means of the finite element method based on the variational approach and simulated using the FreeFEM++ code. A detailed grid-independence study is included. Blood flow, heat, and mass transfer characteristics are examined for the effects of selected geometric, nanoscale, rheological, viscosity, and magnetic parameters, that is, stenotic diameter (d), viscoelastic parameter (λ 1 ), thermophoresis parameter (N t ), Brownian motion parameter (N b ), and magnetic body force parameter (M) at the throat of the stenosis and throughout the arterial domain. The velocity, temperature, and nanoparticle concentration fields are also visualized through instantaneous patterns of contours. An increase in magnetic and thermophoresis parameters is found to enhance the temperature, nanoparticle concentration, and skin-friction coefficient. Increasing Brownian motion parameter is observed to accelerate the blood flow. Narrower stenosis significantly alters the temperature and nanoparticle distributions and magnitudes. The novelty of the study relates to the combination of geometric complexity, multiphysical nanoscale, and thermomagnetic behavior, and also the simultaneous presence of biorheological behavior (all of which arise in actual cardiovascular heat transfer phenomena) in a single work with extensive visualization of the flow, heat, and mass transfer characteristics. The simulations are relevant to the diffusion of nano-drugs in magnettargeted treatment of stenosed arterial disease. K E Y W O R D Sarterial stenosis, finite element method, magnetohydrodynamics, nano-drugs, non-Newtonian blood flow, thermophoresis, Vogel's model

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“…It was introduced for the neutron transport equation by Reed and Hill (1973), analyzed by LaSaint and Raviart (1974) and developed and made popular for fluid dynamics equations by Cockburn, Shu, Bassi, Rebay, and others (see e.g., Bassi and Rebay 1997a,b, Cockburn, Karniadakis, and Shu 2000, Cockburn and Shu 2005, Shu 2012, and the references therein). Efficient DG schemes using the Lagrange polynomials for nodal points as basis functions known as nodal DG methods can be found in (Hesthaven and Warburton 2008).…”

Section: Introductionmentioning

confidence: 98%

High-Order Space-Time Methods for Conservation Laws

Huynh

2013

21st AIAA Computational Fluid Dynamics Conference

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Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a CFL (Courant-Friedrichs-Lewy) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree , then the time step sizes are typically proportional to .) Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown.

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Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments

Cited by 44 publications

References 106 publications

A Non-parametric Discontinuous Galerkin Formulation of the Integral Boundary Layer Equations with Strong Viscous/Inviscid Coupling

A Non-parametric Discontinuous Galerkin Formulation of the Integral Boundary Layer Equations with Strong Viscous/Inviscid Coupling

Finite element analysis of non‐Newtonian magnetohemodynamic flow conveying nanoparticles through a stenosed coronary artery

High-Order Space-Time Methods for Conservation Laws

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