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The paper is a continuation of the efficiency and accuracy analysis of different high order numerical schemes previously published by the authors [1-3]. After the analysis of advection equation and the scalar conservation equation for different initial conditions and for different convex and nonconvex conservative fluxes, we further investigated the one-dimensional conservation equation. This paper focuses on a new comparison of the behavior of the most known Weighted Essentially Non-Oscillatory (WENO) type numerical schemes for shock tube problem. We consider nine different Riemann Problems. Analytical solutions are provided for almost all different initial Riemann conditions. A third-order TVD Runge-Kutta (TVDRK3) scheme was adopted for advancing the solution in time. The motivation of this assessment is related to filling a gap in the specialized literature characterized by the bias towards presenting only cases favorable to a certain method. Our purpose is to present objectively the capacity of each method for solving one-dimensional conservation law problem. Unfortunately, not all the schemes identify accurately the position of the shocks or rarefaction waves and some of them even do not converge to a solution for different Riemann initial conditions. [ABSTRACT FROM AUTHOR]

This paper focuses on a new comparison of the behavior of three Weighted Essentially Non-Oscillatory (WENO) type numerical schemes for three different nonlinear fluxes, in the case of scalar conservation law. The analytical solution is provided for various boundary conditions. For the time integration we adopt the 4-6 stage Low-Dispersion Low-Dissipation Runge-Kutta method (LDDRK 4-6). The schemes were tested on piecewise constant function for non-periodical conditions. The assessment was performed because the specialized literature mainly presents cases favorable illustrating only to a particular method while our purpose is to objectively present the performance and capacity of each method to simulate simple cases like scalar conservation law problems. All the schemes accurately identify the position of the shock and converge to the proper weak solution for the non-linear fluxes and different initial conditions. The paper is a continuation of the efficiency and accuracy analysis of high order numerical schemes previously published by the authors [1,2]. [ABSTRACT FROM AUTHOR]

Supersonic/hypersonic flows with strong shocks need special treatment in Computational Fluid Dynamics (CFD) in order to accurately capture the discontinuity location and his magnitude. To avoid numerical instabilities in the presence of discontinuities, the numerical schemes must generate low dissipation and low dispersion error. Consequently, the algorithms used to calculate the time and space-derivatives, should exhibit a low amplitude and phase error. This paper focuses on the comparison of the numerical results obtained by simulations with some high resolution numerical schemes applied on linear and non-linear one-dimensional conservation low. The analytical solutions are provided for all benchmark tests considering smooth periodical conditions. All the schemes converge to the proper weak solution for linear flux and smooth initial conditions. However, when the flux is non-linear, the discontinuities may develop from smooth initial conditions and the shock must be correctly captured. All the schemes accurately identify the shock position, with the price of the numerical oscillation in the vicinity of the sudden variation. We believe that the identification of this pure numerical behavior, without physical relevance, in 1D case is extremely useful to avoid problems related to the stability and convergence of the solution in the general 3D case. [ABSTRACT FROM AUTHOR]