Several Models for Verificating Numerical Solution of Euler Equation

The authors have successively found numerical examples in which the computational error of the WENO is larger than that of the first-order upwinding scheme under specific conditions since 2018. After qualitative analyses, the authors attribute this phenomenon to the fact that the method of constructing the scheme using a spatial multipoint stencil does not comply with the characteristic theory of hyperbolic equation as well as the introduction of unphysical fluctuations in the flux splitting scheme. This paper was a collection of examples from various comparative numerical experiments conducted over the years based on the Euler equations for the first-order upwinding, MUSCL, and WENO schemes to demonstrate this point of view. It is hoped that it will be used as a verification model for solving Euler equations by the finite difference method for the reference of colleagues. Currently, the classical examples of verification of higher-order schemes in domestic and inter-national literature, such as isentropic vortices, double Mach reflections, shock waves and free-interface interactions, etc., are mostly verified based on the qualitative comparison of numerical phenomena and lack of quantitative indicators. The verifi-cation model in this paper is able to calculate the numerical error, which can be evaluated quantitatively. By analyzing these validation models, an algorithm was proposed that can effectively reduce the induced error of the initial shock wave.