Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires

Alouges, François; Vuyst, Florian De; Coq, G. Le; Lorin, Emmanuel

doi:10.1016/s1631-073x(02)02514-1

Cited by 8 publications

(8 citation statements)

References 5 publications

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“…In this paper we have presented an extension of the reservoir technique [2,3] in order to limit the numerical diffusion of finite volume schemes approaching advection equations, in the two-dimensional framework. The main idea, compared to the one-dimensional case is the combination of the correction and reservoir techniques.…”

Section: Resultsmentioning

confidence: 99%

“…To this end we introduce a local time counter denoted by c n (K) for all K in T 0 (Ω), with n denoting the time index. This counter is very close to the one introduced in the one-dimensional framework [2,3]. When the counter reaches the value Δt n K , we update the solution in "emptying-up" the value u 0 K located for instance in L at time t n , in a neighboring cell of L.…”

Section: Reservoir Technique For One-directional Upwinding For Two-dimentioning

confidence: 92%

“…Furthermore, continuous systems self-properties (nonlinearity, entropy, conservation) are often more difficult to obtain for highorder schemes compared to first-order ones. This is why, we propose a complete different approach based on the reservoir technique [1][2][3]. This technique allows us to make Godunov-type schemes very low-diffusive.…”

Section: Introductionmentioning

confidence: 99%

“…As in [1], the idea is to use first-order schemes at high-CFL numbers for each cell and each time step. In [1,3,16], we showed that in the onedimensional case, the "good" properties of first-order schemes (entropy, nonlinearity, conservation, etc) are conserved. Recently many authors have treated the problem of nondiffusive finite volume and finite difference schemes for linear or nonlinear hyperbolic systems.…”

Section: Introductionmentioning

confidence: 99%

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Two-Dimensional Extension of the Reservoir Technique for Some Linear Advection Systems

2006

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In this paper we present an extension of the reservoir technique (see, [Alouges et al., Submitted; Alouges et al.(2002a), In: Finite volumes for complex applications, III, pp. 247-254, Marseille;Alouges et al.(2002b), C. R. Math. Acad. Sci. Paris, 335(7), 627-632.]) for two-dimensional advection equations with non-constant velocities. The purpose of this work is to make decrease the numerical diffusion of finite volume schemes, correcting the numerical directions of propagation, using a so-called corrector vector combined with the reservoirs. We then introduce an object called velocities rose in order to minimize the algorithmic complexity of this method.

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Section: Resultsmentioning

confidence: 99%

Section: Reservoir Technique For One-directional Upwinding For Two-dimentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two-Dimensional Extension of the Reservoir Technique for Some Linear Advection Systems

2006

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“…Let us note that it is easy to prove that when the initial condition is a Heaviside function, the 1D scheme (73)-(75) is equivalent to the reservoir scheme proposed in [3]. The advantage of the formulation (73)-(75) is that it enters into the field of finite-volume schemes (which allows to obtain convergence results with classical techniques) and that the 2D (or 3D) extension is immediate through the splitting technique described in Section 3.4.1.…”

Section: Discretization Of O T Y + Vo X Y = 0 With V :¼ C St With a Nmentioning

confidence: 99%

Numerical resolution of a potential diphasic low Mach number system

Dellacherie

2007

Journal of Computational Physics

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Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems

Fillion‐Gourdeau

Lorin

2018

Numer Algor

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This paper is devoted to the derivation of a digital quantum algorithm for the Cauchy problem for symmetric first order linear hyperbolic systems, thanks to the reservoir technique. The reservoir technique is a method designed to avoid artificial diffusion generated by first order finite volume methods approximating hyperbolic systems of conservation laws. For some class of hyperbolic systems, namely those with constant matrices in several dimensions, we show that the combination of i) the reservoir method and ii) the alternate direction iteration operator splitting approximation, allows for the derivation of algorithms only based on simple unitary transformations, thus perfectly suitable for an implementation on a quantum computer. The same approach can also be adapted to scalar one-dimensional systems with non-constant velocity by combining with a non-uniform mesh. The asymptotic computational complexity for the time evolution is determined and it is demonstrated that the quantum algorithm is more efficient than the classical version. However, in the quantum case, the solution is encoded in probability amplitudes of the quantum register. As a consequence, as with other similar quantum algorithms, a post-processing mechanism has to be used to obtain general properties of the solution because a direct reading cannot be performed as efficiently as the time evolution.Keywords First order hyperbolic systems, quantum algorithms, quantum information theory, reservoir method. IntroductionQuantum computing is a new paradigm in information science which benefits from quantum mechanics to perform some computational tasks. In the last few decades, it has attracted a lot of attention because it promises efficient solutions to a large class of problems deemed unsolvable on classical computers. Shor's algorithm, for the prime number factorization of integers, is the foremost example of the strength of quantum computing [52]. This algorithm runs in polynomial time, i.e. the computation time scales like a polynomial function of the input size, while the same task runs in (sub-)exponential time on a classical computer. This quantum speedup has motivated the development of many other algorithms for the solution of problems in the BQP complexity class but outside the P class, i.e. problems with bounded error for which the amount of quantum resources is a polynomial function and having an exponential speedup over classical computations [47,56].One of the promising applications of quantum computing is the simulation of quantum systems. Inspired from Feynman's quantum simulator [24], it has been demonstrated that universal quantum F. Fillion-Gourdeau

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Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires

Cited by 8 publications

References 5 publications

Two-Dimensional Extension of the Reservoir Technique for Some Linear Advection Systems

Two-Dimensional Extension of the Reservoir Technique for Some Linear Advection Systems

Numerical resolution of a potential diphasic low Mach number system

Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems

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