On difference schemes of third order accuracy for nonlinear hyperbolic systems

Rusanov, V. V.

doi:10.1016/0021-9991(70)90077-x

Cited by 98 publications

(33 citation statements)

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“…The MHD equations have been solved by using the explicit conservative Rusanov shock-capturing scheme of third order of accuracy [Rusanov, 1970]. The numerical scheme has been tested by the one-dimensional auto-model hydrodynamic problem.…”

Section: The Anisotropic Double Adiabatic Mhd Model (Or Thementioning

confidence: 99%

Application of the bounded anisotropy model for the dayside magnetosheath

Samsonov¹,

Pudovkin²

2000

J. Geophys. Res.

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Section: The Anisotropic Double Adiabatic Mhd Model (Or Thementioning

confidence: 99%

Application of the bounded anisotropy model for the dayside magnetosheath

Samsonov¹,

Pudovkin²

2000

J. Geophys. Res.

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“…Using the staggered mesh, one gets in 1 space dimensions the finite difference scheme (3.20)-(3.22) without the vector H; it is then stable provided that Xp(/1) ^ f. In the 2 space dimension case, we obtain the sufficient condition of Xpm;ix ¿¡ \. The scheme proposed by Rousanov [10] is also essentially the one given by Eqs. Had we chosen to also develop explicitly the finite difference schemes which are implied by Theorem 2 (i.e., utilize the coefficient matrices A, as well as the conservation vectors F¡), then we immediately get, for the 2nd order case, the Lax-Wendroff scheme [6] and, for the 3rd order case, a scheme considered earlier by the present authors (unpublished).…”

Section: ¿-Imentioning

confidence: 97%

“…361] and Strang [4], [8]. Burstein and Mirin [9] and Rusanov [10] then solved the 3rd order accuracy case while Strang's [4] work included arbitrary order of accuracy for a linear system in one space dimension.…”

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confidence: 99%

Higher Order Accuracy Finite Difference Algorithms for Quasi-Linear, Conservation Law Hyperbolic Systems

Abarbanel¹,

Gottlieb²

1973

Mathematics of Computation

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Abstract. An explicit algorithm that yields finite difference schemes of any desired order of accuracy for solving quasi-linear hyperbolic systems of partial differential equations in several space dimensions is presented. These schemes are shown to be stable under certain conditions. The stability conditions in the one-dimensional case are derived for any order of accuracy. Analytic stability proofs for two and d (d > 2) space dimensions are also obtained up to and including third order accuracy. A conjecture is submitted for the highest accuracy schemes in the multi-dimensional cases. Numerical examples show that the above schemes have the stipulated accuracy and stability.Introduction. The task of solving numerically the equations of gas dynamics has given rise in the last 25 years to the search for finite difference algorithms of increasing accuracy and efficiency. In the present paper, the following results are presented:(1) An explicit algorithm that yields finite difference equations that approximate the quasi-linear hyperbolic system to any desired accuracy and for arbitrary number of space dimensions.(2) Analytic stability proofs and criteria of the above-mentioned algorithms in the case of one dimension, for arbitrary order of accuracy.(3) Analytic stability proofs in the 2 and d id > 2) dimensional cases up to and including 3rd order accuracy with sufficient stability conditions.(4) Numerical examples are carried out for a one-dimensional 2X2 system and a two-dimensional 2X2 system. The computed values are compared with analytic solutions and are shown to have the stipulated accuracies (4th order for the 1-D case and 3rd order for the 2-D case). "

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“…Theorem 1. Under the condition of (14) the SHASTA FCT algorithm is L°°-stable, and it holds that (19) inf P0(*) < Pj < SUP Po(x) for any j and nonnegative k.…”

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confidence: 99%

On the SHASTA FCT algorithm for the equation ∂𝜌/∂𝑡+(∂/∂𝑥)(𝑣(𝜌)𝜌)=0

Ikeda¹,

Nakagawa²

1979

Math. Comp.

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Abstract.In recent years, Boris, Book and Hain have proposed a family of finite difference methods called FCT techniques for the Cauchy problem of the continuity equation. The purpose of this paper is to study the stability and convergence about the SHASTA FCT algorithm, which is one of the basic schemes among many FCT techniques, though not in its original form but a slightly modified one for our technical reason.

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On difference schemes of third order accuracy for nonlinear hyperbolic systems

Cited by 98 publications

References 0 publications

Application of the bounded anisotropy model for the dayside magnetosheath

Application of the bounded anisotropy model for the dayside magnetosheath

Higher Order Accuracy Finite Difference Algorithms for Quasi-Linear, Conservation Law Hyperbolic Systems

On the SHASTA FCT algorithm for the equation ∂𝜌/∂𝑡+(∂/∂𝑥)(𝑣(𝜌)𝜌)=0

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