Simplified Second-Order Godunov-Type Methods

Davis, Stephen F.

doi:10.1137/0909030

Cited by 389 publications

(214 citation statements)

References 17 publications

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“…One possible choice is to set S L = λ 1,i+1/2 and S R = λ N,i+1/2 , although some other different possibilities have been proposed in the bibliography, as, for example (see Davis [11])…”

Section: Remarkmentioning

confidence: 99%

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

Díáz¹,

Fernández-Nieto²

2012

SIAM J. Sci. Comput.

108

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Abstract. In this work, we present a class of fast first order finite volume solvers, named as PVM (Polynomial Viscosity Matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of Roe Matrix is needed. As consequence, they are faster than Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [12] for balance laws and nonconservative systems. The first-order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multi-dimensional problems. In this work, some well known solvers as Rusanov, FORCE (see [30], [8]), GFORCE (see [31], [8]) or HLL (see [18]) are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented and the performance of the numerical schemes are compared among them and with Roe scheme.

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“…One possible choice is to set S L = λ 1,i+1/2 and S R = λ N,i+1/2 , although some other different possibilities have been proposed in the bibliography, as, for example (see Davis [11])…”

Section: Remarkmentioning

confidence: 99%

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

Díáz¹,

Fernández-Nieto²

2012

SIAM J. Sci. Comput.

108

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“…Various choices are possible for the wave speed estimates; see for example [34]. Here we use Davis estimates [7]:…”

Section: Extension To Other Schemesmentioning

confidence: 99%

A Simple Method for Compressible Multifluid Flows

Saurel¹,

Abgrall²

1999

SIAM J. Sci. Comput.

371

346

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Abstract. A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunov-type scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 10 5 ). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer-Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes.

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“…where: wave speeds, S L and S R (L and R subscripts denote the left and right states, respectively), for air flows, employ the direct wave speed estimation, as suggested by Davis (1988), which is the most well-known approach for estimating bounds for the minimum and maximum signal velocities in the solution of the Riemann problem:…”

Section: Methodsmentioning

confidence: 99%

A Computational Modeling for Semi-Coupled Multiphase Flow in Atmospheric Icing Conditions

Jung

2015

J.Aerosp. Technol. Manag.

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A two-dimensional second-order positivitypreserving finite volume upwind scheme is developed for a semi-coupled algorithm involving the air and droplet flow fields in the Eulerian frame, which shares the grid for each phase. Special emphasis is placed on the computational modeling, which is induced from a strongly-coupled algorithm that satisfies the strict hyperbolicity and its numerical scheme based on the Harten-Lax-van Leer-Contact solver preserving the positivity to handle multiphase flow in the Eulerian frame. The proposed modeling associated with the semi-coupled algorithm including the Navier-Stokes and droplet equations takes into account different boundary conditions on the solid surface for each phase. The verification and validation studies show that the new scheme can solve the air and droplet flow fields in fairly good agreement with the exact analytical solutions and experimental data. In particular, it accurately predicted the maximum value of the droplet impingement intensity near the stagnation region and the droplet impingement area.

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Simplified Second-Order Godunov-Type Methods

Cited by 389 publications

References 17 publications

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

A Simple Method for Compressible Multifluid Flows

A Computational Modeling for Semi-Coupled Multiphase Flow in Atmospheric Icing Conditions

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