Classification of difference schemes of gas dynamics by the method of differential approximation—I. One-dimensional case

Yanenko, N. N.; Fedotova, Z. I.; Kompaniets, L. A.; Shokin, Yu. I.

doi:10.1016/0045-7930(83)90030-0

Cited by 15 publications

(5 citation statements)

References 12 publications

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“…Inserting the numerical solution (7) in this scheme results in w r = 1 t arcsin(C 0 sin(k x)); w i = 0; (13) and the numerical (deterministic) leapfrog solution becomes c n j = sin(kx j ? vm l t n ) (14) with m l = 1 C 0 x arcsin(C 0 sin(k x)): (15) No arti cial damping is introduced, and the accuracy of the phase depends on how close m l approximates the wavenumber k. For a normally distributed velocity the moments of c n j may be derived. The rst two moments become E c n j = exp ?…”

mentioning

confidence: 99%

A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

Osnes¹,

Langtangen²

1998

SIAM J. Sci. Comput.

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We study a uni-directional hyperbolic transport equation, with a homogeneous stochastic transport velocity, solved by Monte Carlo simulation. Several nite di erence schemes are applied to the deterministic problem in each Monte Carlo iteration, and the numerical solution of the stochastic problem is compared to analytical solutions derived in the paper. We present both a theoretical analysis and summarized results from extensive numerical experiments. The behavior of the various schemes depend on the stochastic properties of the problem, and there are new demands to the schemes when they are used as part of a Monte Carlo simulation. For example, schemes that are very oscillatory for a single deterministic problem, like the leapfrog scheme, turn out to be e cient and accurate for the corresponding stochastic problem.

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mentioning

confidence: 99%

A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

Osnes¹,

Langtangen²

1998

SIAM J. Sci. Comput.

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“…This analysis can be best performed by constructing the EDE of the difference scheme. The traditional approach for constructing EDEs is to replace the terms in the difference equation by their Taylor expansions (O'Brian et al 1951;Yanenko et al 1983;Leonard 1979;Noye 1991). The associated solution of the EDE is a continuous function that is equal to the solution of the difference equation at all grid nodes.…”

Section: Equivalent Differential Equationsmentioning

confidence: 99%

“…For example, if these extra terms tend to zero as the grid size tends to zero, then the finite-difference equation is consistent with the physical model. Further, if the governing equation is used to transform the nonphysical temporal derivatives into spatial derivatives, the presence of even order derivatives implies that the numerical scheme is dissipative, whereas odd order derivatives imply that the scheme is dispersive (Yanenko et al 1983;Leonard 1979;Noye 1991).…”

Section: Equivalent Differential Equationsmentioning

confidence: 99%

Equivalent Differential Equations in Fixed‐Grid Characteristics Method

Ghidaoui

Karney

1994

J. Hydraul. Eng.

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Various numerical techniques, such as wave-speed adjustment or interpolation, are generally required in order to apply the fixed-grid method of characteristics to multipipe systems or to systems with variable wave speed. However, these techniques introduce into the solution unwanted side effects such as numerical attenuation and dispersion. The present paper develops the concept of an equivalent hyperbolic differential equation to study how discretization errors arise in pipeline applications for the most common interpolation techniques. In particular, it is shown that space-line interpolation and the Holly-Preissmann scheme are equivalent to a wave-diffusion model with an adjusted wave speed, but that the latter method has additional source and sink terms. Further, time-line interpolation is shown to be equivalent to a superposition of two waves with different wave speeds. In general, the equivalent hyperbolic differential equation concept evaluates the consistency of the numerical scheme, provides a mathematical description of the numerical dissipation and dispersion, gives an independent way of determining the Courant condition, allows the comparison of alternative approaches, finds the wave path, and explains why higher-order methods should usually be avoided.

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“…Two different methods for incorporating symmetry concepts into the discretization of differential equations exist in the literature. One was proposed and explored by Shokin and Yanenko [4][5][6][7][8] for PDE's and has been implemented in several recent studies [9,10]. It is called the differential approximation method and the basic idea is the following.…”

Section: Introductionmentioning

confidence: 99%

“…One was proposed and explored by Shokin and Yanenko [4][5][6][7][8] for PDEs and has been implemented in several recent studies [9,10]. It is called the differential approximation method and the basic idea is the following.…”

Section: Introductionmentioning

confidence: 99%

Lie Groups and Numerical Solutions of Differential Equations: Invariant Discretization Versus Differential Approximation

Levi

Winternitz

2013

Acta Polytech

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Classification of difference schemes of gas dynamics by the method of differential approximation—I. One-dimensional case

Cited by 15 publications

References 12 publications

A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

Equivalent Differential Equations in Fixed‐Grid Characteristics Method

Lie Groups and Numerical Solutions of Differential Equations: Invariant Discretization Versus Differential Approximation

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