Improvement of multistep WENO scheme and its extension to higher orders of accuracy. (6th May 2016) - Ma, Yankai; Yan, Zhenguo; Zhu, Huajun
Summary: This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non‐oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth‐order accurate multistep methods are devised, providing weighted essentially non‐oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non‐oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock‐capturing problems. Copyright © 2016 John Wiley & Sons, Ltd. Abstract : A new class of multistep WENO methods is presented through using new modified nonlinear weights. The weights definition takes into account the novel extra information on the regularity of the solution and renders smoothness indicators closer to uniformity so as to increase the resolution power when approximating smooth solutions. This new method provides WENO schemes with enhanced order of convergence at transition points while maintaining stability and the ENO behavior.