Spectral/hp Methods for Viscous Compressible Flows on Unstructured 2D Meshes

Lomtev, I.; Quillen, C. B.; Karniadakis, George Em

doi:10.1006/jcph.1997.5831

Cited by 77 publications

(69 citation statements)

References 33 publications

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“…See also the review in [35]. Finally, Lomtev, Quillen, and Karniadakis [57] used the DG-space discretization method to handle the convective part of the compressible Navier-Stokes equations combined with a mixed method to approximate the diffusive part of the equations. All the above methods used discontinuous approximation only for the convective terms and mixed methods for second-order elliptic operators.…”

Section: 3mentioning

confidence: 99%

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

Arnold

Brezzi

Cockburn

et al. 2002

SIAM J. Numer. Anal.

2,803

2,811

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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems. . These DG methods were then usually called interior penalty (IP) methods, and their development remained independent of the development of the DG methods for hyperbolic equations. In this paper, we present a detailed study of a class of DG methods for second-order elliptic problems which includes all the above-mentioned methods.Next, we introduce the DG methods. For the sake of simplicity and easy presentation of the main ideas, we restrict ourselves to the model problem

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Section: 3mentioning

confidence: 99%

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

Arnold

Brezzi

Cockburn

et al. 2002

SIAM J. Numer. Anal.

2,803

2,811

View full text Add to dashboard Cite

show abstract

“…The discontinuous Galerkin formulation is a convenient formulation for high-order discretisation of hyperbolic conservation laws. Following the work of Cockburn and Shu [24] and Lomtev, Quillen and Karniadakis [25] we proceed as follows.…”

Section: Numerical Discretisationmentioning

confidence: 99%

One-dimensional modelling of a vascular network in space-time variables

Sherwin

Franke

Peiró

et al. 2003

Journal of Engineering Mathematics

398

497

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Abstract. In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled.In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A, Q), area-velocity (A, u), pressure-velocity (p, u) and pressure-mass flux(p, Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A, u) variables the extension of the single vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves.The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second order Adams-Bashforth time integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)).Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.

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“…This discontinuous basis is orthogonal, hierarchical, and maintains a tensor-product property (even for tetrahedra) [3]. In a previous work [4], a hybrid method based on a discontinuous Galerkin formulation for the hyperbolic contribution and a mixed Galerkin formulation for the diffusive contribution was developed. Such a formulation requires two sets of trial basis, one in L 2 and another one in C 0 .…”

Section: Introductionmentioning

confidence: 99%