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Computational fluid dynamics simulations using the WENO-LF method are applied to high Mach number nonrelativistic astrophysical jets, including the effects of radiative cooling. Our numerical methods have allowed us to simulate astrophysical jets at much higher Mach numbers than have been attained (Mach 20) in the literature. Our simulations of the HH 1-2 astrophysical jets are at Mach 80. Simulations at high Mach numbers and with radiative cooling are essential for achieving detailed agreement with the astrophysical images.

This paper is concerned with fifth-order weighted essentially non-oscillatory (WENO) scheme with a new smoothness indicator. As the so-called WENO-JS scheme (Jiang and Shu in J Comput Phys 126:202-228, 1996) provides the third-order accuracy at critical points where the first and third order derivatives do not becomes zero simultaneously, several fifth-order WENO scheme have been developed through modifying the known smoothness indicators of WENO-JS. Recently a smoothness indicator based on L 1 -norm has been proposed by Ha et al. (J Comput Phys 232:68-86, 2013) (denoted by WENO-NS). The aim of this paper is twofold. Firstly, we further improve the smoothness indicator of WENO-NS and secondly, using this measurement, we suggest new nonlinear weights by simplifying WENO-NS weights. The proposed WENO scheme provides the fifth-order accuracy, even at critical points. Some numerical experiments are provided to demonstrate that the present scheme performs better than other WENO schemes of the same order.

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The aim of this study is to develop a novel sixth-order weighted essentially non-oscillatory (WENO) finite difference scheme. To design new WENO weights, we present two important measurements: a discontinuity detector (at the cell boundary) and a smoothness indicator. The interpolation method is implemented by using exponential polynomials with tension parameters such that they can be tuned to the characteristics of the given data, yielding better approximation near steep gradients without spurious oscillations, compared to the WENO schemes based on algebraic polynomials at lower computational cost. A detailed analysis is performed to verify that the proposed scheme provides the required convergence order of accuracy. Some numerical experiments are presented and compared with other sixth-order WENO schemes to demonstrate the new algorithm's ability.

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High Mach number astrophysical jets are simulated using a positive scheme, and are compared with WENO-LF simulations. A version of the positive scheme has allowed us to simulate astrophysical jets with radiative cooling up to Mach number 270 with respect to the heavy jet gas, a factor of two times higher than the maximum Mach number attained with the WENO schemes and ten times higher than with CLAWPACK. Such high Mach numbers occur in many settings in astrophysical flows, so it is important to develop a scheme that can simulate at these Mach numbers.

The nonnegative fundamental solution of a scalar conservation law is constructed when its flux may have a finite number of inflection points. The constructed solution can be either explicit and implicit depending on the flux. This fundamental solution consists of a series of rarefaction waves, contact discontinuities and a shock. These analytically constructed fundamental solutions are also compared with numerical approximations, which possess the structure of the analytically constructed fundamental solution.