Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics

Kolgan, V. P.

doi:10.1016/j.jcp.2010.12.033

Cited by 108 publications

(93 citation statements)

References 6 publications

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“…The update requires numerical fluxes, defined in terms of exact or approximate solutions of Riemann problems (along the normal direction) at each cell interface. Higher-order spatial accuracy is recovered by employing non-oscillatory piecewise polynomial reconstructions like secondorder TVD [26,27,50], higher-order ENO [23] and WENO [40] methods. Higher-order temporal accuracy results from using stability-preserving Runge-Kutta methods [22].…”

Section: Survey Of Available Numerical Methodsmentioning

confidence: 99%

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Fuchs

McMurry

Mishra

et al. 2011

Commun. comput. phys.

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Abstract. We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three-and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO-and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first-and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

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Section: Survey Of Available Numerical Methodsmentioning

confidence: 99%

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Fuchs

McMurry

Mishra

et al. 2011

Commun. comput. phys.

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“…Such development effort was undertaken by Kolgan [3] who proposed to suppress spurious oscillations and produced in this way a non-oscillatory Godunovtype scheme of second order spatial accuracy. Further, more well-known, developments were due to van Leer [4] who extended Godunov's approach to second-order spatial accuracy by the MUSCL approach.…”

Section: Muscl Methodmentioning

confidence: 99%

“…By combining schemes with predominantly both positive and negative phase errors, Fromm received low dispersive errors [2]. Later on Kolgan [3] proposed to reduce spurious oscillations by applying the socalled principle of minimal values of derivatives, producing in this manner a non-oscillatory Godunov-type scheme of second order spatial accuracy. Van Leer [4] developed Monotone Upstream Scheme for Conservation Laws (MUSCL) in which he included a linear representation of a solution within each numerical cell.…”

Section: Introductionmentioning

confidence: 99%

Implementation of MUSCL-Hancock method into the C++ code for the Euler equations

Murawski¹,

Stpiczyński²

2012

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. In this paper we present implementation of the MUSCL-Hancock method for numerical solutions of the Euler equations. As a result of the internal complexity of these equations solving them numerically is a formidable task. With the use of the original C++ code, we developed and presented results of a numerical test that was performed. This test shows that our code copes very well with this task.

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“…The history of the explosion theory is in detail expound [3][4]. Development of finite-difference numerical methods [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] have allowed to simulate discontinuous solutions of the equations of inviscid gas dynamics. Currently, the problem of the explosion is considered as a test for modern numerical schemes of a high order of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Instability of Contact Surface in Cylindrical Explosive Waves

Tunik¹

2017

Fluid Mech Open Acc

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In this paper are developed modifications of the Godunov scheme, based on Kolgan's scheme of the second order of accuracy in the spatial variables for smooth solutions. It is constructed schemes of the first and the variable order of approximation, which exceed the Godunov scheme in accuracy. Referencing to the system of differential equations for propagation of flat sound waves in a gas at rest, the Kolgan scheme and the first-order schemes obtained are investigated onto the ability to ensure the non decrease of entropy, that is, to product of physically justified numerical solutions.The test problems of nonlinear gas dynamics on the decay of a discontinuity in a pipe and the transformation of a non uniformity in a plane-parallel flow are solved. Cylindrical explosion task is considered as the main one. The stability of a contact discontinuity behind a blast wave is investigated numerically in the Cartesian and polar coordinate systems. Analysis of obtained and published solutions does not confirm the instability of the contact discontinuity which initially has the circular shape. Change of the shape of initially perturbed break is largely caused by the instability of Taylor, not Richtmyer-Meshkov. Calculations are partially fulfilled using supercomputer "Lomonosov" of Moscow state university.

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Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics

Cited by 108 publications

References 6 publications

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Implementation of MUSCL-Hancock method into the C++ code for the Euler equations

Instability of Contact Surface in Cylindrical Explosive Waves

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