Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution (Communications in Computational Physics, 2020)
The method of mapping function was first proposed by Henrick et al. [J.Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth-orderWENO scheme, and through which the requirement of convergence order issatisfied and the performance of the scheme is improved. Different from Henrick'smethod, a concept of piecewise polynomial function is proposed in this study andcorresponding WENO schemes are obtained. The advantage of the new method isthat the function can have a gentle profile at the location of the linear weight (or themapped nonlinear weight can be close to its linear counterpart), and therefore is favorablefor the resolution enhancement. Besides, the function also has the flexibilityof quick convergence to identity mapping near two endpoints of [0,1], which is favorablefor improved numerical stability. The fourth-, fifth- and sixth-order polynomialfunctions are constructed correspondingly with different emphasis on aforementionedflatness and convergence. Among them, the fifth-order version has the flattest profile.To check the performance of the methods, the 1-D Shu-Osher problem, the 2-DRiemann problem and the double Mach reflection are tested with the comparison ofWENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolutionfor describing shear-layer instability of the Riemann problem, and they alsoindicate high resolution in computations of double Mach reflection, where only theseproposed schemes successfully resolved the vortex-pairing phenomenon. Other investigationshave shown that the single polynomial mapping function has no advantageover the proposed piecewise one, and it is of no evident benefit to use the proposedmethod for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomialand corresponding WENO scheme are suggested for resolution improvement.