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A conservative spectral element method for the approximation of compressible fluid flow

Kelly Black (1999)

Kybernetika

A method to approximate the Euler equations is presented. The method is a multi-domain approximation, and a variational form of the Euler equations is found by making use of the divergence theorem. The method is similar to that of the Discontinuous-Galerkin method of Cockburn and Shu, but the implementation is constructed through a spectral, multi-domain approach. The method is introduced and is shown to be a conservative scheme. A numerical example is given for the expanding flow around a point...

A modified Cayley transform for the discretized Navier-Stokes equations

K. A. Cliffe, T. J. Garratt, Alastair Spence (1993)

Applications of Mathematics

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...

A new domain decomposition method for the compressible Euler equations

Victorita Dolean, Frédéric Nataf (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains,...

A numerical method for unsteady flows

Nicola Botta, Rolf Jeltsch (1995)

Applications of Mathematics

A high resolution finite volume method for the computation of unsteady solutions of the Euler equations in two space dimensions is presented and validated. The scheme is of Godunov-type. The first order part of the flux function uses the approximate Riemann problem solver of Pandolfi and here a new derivation of this solver is presented. This construction paves the way to understand the conditions under which the scheme satisfies an entropy condition. The extension to higher order is done by applying...

A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil

François Beux, Maria-Vittoria Salvetti, Alexey Ignatyev, Ding Li, Charles Merkle, Edoardo Sinibaldi (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The results of a workshop concerning the numerical simulation of the liquid flow around a hydrofoil in non-cavitating and cavitating conditions are presented. This workshop was part of the conference “Mathematical and Numerical aspects of Low Mach Number Flows” (2004) and was aimed to investigate the capabilities of different compressible flow solvers for the low Mach number regime and for flows in which incompressible and supersonic regions are simultaneously present. Different physical models...

A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil

François Beux, Maria-Vittoria Salvetti, Alexey Ignatyev, Ding Li, Charles Merkle, Edoardo Sinibaldi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The results of a workshop concerning the numerical simulation of the liquid flow around a hydrofoil in non-cavitating and cavitating conditions are presented. This workshop was part of the conference “Mathematical and Numerical aspects of Low Mach Number Flows” (2004) and was aimed to investigate the capabilities of different compressible flow solvers for the low Mach number regime and for flows in which incompressible and supersonic regions are simultaneously present. Different physical models...

A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Mario Ohlberger (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation c t + · ( 𝐮 f ( c ) ) - · ( D c ) + λ c = 0 . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1 -norm, independent of the diffusion parameter D . The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Mario Ohlberger (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation c t + · ( 𝐮 f ( c ) ) - · ( D c ) + λ c = 0 . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L1-norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

A second-order multi-fluid model for evaporating sprays

Guillaume Dufour, Philippe Villedieu (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. Using the underlying kinetic interpretation of the sectional method [Y. Tambour, Combust. Flame 60 (1985) 15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model. 5 (2001) 537–572], we propose an extension of this approach based on a more accurate representation of the droplet size number density in each...

A second-order multi-fluid model for evaporating sprays

Guillaume Dufour, Philippe Villedieu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. Using the underlying kinetic interpretation of the sectional method [Y. Tambour, Combust. Flame60 (1985) 15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model.5 (2001) 537–572], we propose an extension of this approach based on a more accurate representation of the droplet size number density in each...

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