Power ENO methods: A fifth-order accurate weighted power ENO method.
A new class of parabolic essentially non-oscillatory methods (\textit{power ENO methods}) is introduced for numerical approximation to nonlinear conservation laws. These methods are based on a class of limiters that contains \textit{minmod} and ENO limiters as a particular case, as well as the limiters based on the harmonic mean, used to design the PHM methods and harmonic ENO methods. The study is focused on ENO \textit{parabolas and the limiters on neighboring second-order differences} is applied in order to retain more information of fine scales. The main advantage of those methods with respect to the classical ENO methods is the improved behavior near discontinuities. In particular, a new fifth-order accurate weighted ENO method is proposed with a better behavior near jumps and corners than the WENO5 method of \textit{G. S. Jiang} and \textit{C. W. Shu} [J. Comput. Phys. 126, No. 1, 202--228 (1996; Zbl 0877.65065)]. A disadvantage of applying limiters on second-order differences is that there is a loss of accuracy near smooth inflection points. In numerical experiments where fine structures appear to be important (e. g., vortex regions), the weighted power ENO method shows substantial improvements in resolving fine scales, in spite of the improvements observed in standard numerical tests might seem to be minor. The paper is organized as follows: In Section 2 the new family of limiters is considered. Section 3 is focused on the new power ENO method: the advantages and disadvantages are discussed. The method is compared to the parabolic ENO method. In particular, the third-order accurate power ENO3 method is discussed. In Section 4, a new fifth-order accurate weighted power ENO method is described. This method is designed as a convex combination of the three parabolas used to define the third-order accurate Power ENO method, following a similar procedure to the one used by Jiang and Shu [loc. cit.]. In Section 5, the power ENO and weighted power ENO methods are tested on several one- and two-dimensional model problems, for scalar and systems of conservation laws. These methods are compared with ENO3 and WENO5 schemes, to analyze their ability in resolving shocks and complex flow problems. The conclusions are summarized in Section 6.
scientific article; zbMATH DE number 2056056