109 research outputs found

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    Riemann Invariant Manifolds for the Multidimensional Euler Equations Part I: Theoretical Development Part II: A Multidimensional Godunov Scheme and Applications

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    A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. The general case of the unsteady compressible Euler equations in several space dimensions is examined. A family of spacetime manifolds is found on which an equivalent one-dimensional problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann Invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann Invariant Manifolds (RIM). In this special case of zero entropy gradients, the equations of motion are integrable on these manifolds, and the problem of computing the solution becomes that of determining the manifold geometry in spacetime. This situation is completely to the traditional method of characteristics in one-dimensional flow. Explicit espressions for the local differential geometry of these manifolds can be found directly from the equations of motion. The local direction and speed of propagation of the waves that these manifolds represent, can be found as a function of the local spatial gradients of the flow. Their geometry is examined, and in particular, their relation to the characteristic surfaces. It turns out that they can be space-like or time-like, depending on the flow gradients. Wave propagation can be viewed as a superposition of these Riemann Invariant waves, whenever appropriate conditions of smoothness are met. This provides a means for decomposing the equations into a set of convective scalar fields in a way which is different and potentially more useful than the characteristic decomposition. The two decompositions become identical in the special case of one-dimellsional flow. This different approach can be used for computational purposes by discretizing the equivalent set of scalar equations. Such a computational application of this theory leads to the possibility of determining the solution at points in spacetime using information that propagates faster than the local characteristic speed, i.e., using information outside the domain of dependence. This possibility and its relation to the uniqueness theorems is discussed

    Time-Independent Gravitational Fields in the BGK Scheme for Hydrodynamics

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    We incorporate a time-independent gravitational field into the BGK scheme for numerical hydrodynamics. In the BGK scheme the gas evolves via an approximation to the collisional Boltzmann equation, namely the Bhatnagar-Gross-Krook (BGK) equation. Time-dependent hydrodynamical fluxes are computed from local solutions of the BGK equation. By accounting for particle collisions, the fundamental mechanism for generating dissipation in gas flow, a scheme based on the BGK equation gives solutions to the Navier-Stokes equations: the fluxes carry both advective and dissipative terms. We perform numerical experiments in both 1D Cartesian geometries and axisymmetric cylindrical coordinates.Comment: 31 pages including 19 figures (For higher resolution figs. see http://www.mpia-hd.mpg.de/MPIA/Projects/THEORY/slyz), Accepted for publication in Astronomy and Astrophysics, Supplement Serie

    Numerical modeling of two-phase flows using the two-fluid two-pressure approach

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    The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity twopressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity for the definition of Rankine-Hugoniot jump relations. Each field of the convective system is investigated, providing that the maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two Finite Volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows

    Numerical modeling of two-phase flows using the two-fluid two-pressure approach

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    The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity twopressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity for the definition of Rankine-Hugoniot jump relations. Each field of the convective system is investigated, providing that the maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two Finite Volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows

    Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

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    We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

    High-accurate SPH method with Multidimensional Optimal Order Detection limiting

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    International audienceWe present a new high-accurate, stable and low-dissipative Smooth Particle Hydrodynamics (SPH) method based on Riemann solvers. The method derives from the SPH-ALE formulation first proposed by Vila and Ben Moussa. Moving Least Squares approximations are used for the reconstruction of the variables and the computation of Taylor expansions. The stability of the scheme is achieved by the a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm. Such a procedure enables to provide genuine gains in accuracy both for one-and two-dimensional problems involving non-smooth flows when compared to classical SPH methods
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