An adaptive characteristic-wise reconstruction WENO-Z scheme for gas dynamic euler equations

Abstract

Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much higher computational cost and thus is not suitable for large scale simulation such as direct numerical simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO-Z scheme, i.e. the AdaWENO-Z scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. By defining shared smoothness functions, shared smoothness indicators as well as shared WENO weights are firstly introduced to reduce the computational cost of the componentwise reconstruction procedure and to define a global switch function capable of detecting discontinuity. According to the given switch function, the new scheme performs characteristic-wise reconstruction near discontinuities and switches to component-wise reconstruction for smooth regions. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO-Z scheme. Numerical results show that AdaWENO-Z maintains essentially non-oscillatory flow field near discontinuities as with the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction method, AdaWENO-Z is about 20% to 40% faster which indicates its excellent efficiency.

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Arbitrary high‐order extended essentially non‐oscillatory schemes for hyperbolic conservation laws

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