Multistep weighted essentially non‐oscillatory scheme

SUMMARYThis paper presents an efficient procedure for overcoming the deficiency of weighted essentially nonoscillatory 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.

Section: Numerical Experimentsmentioning

confidence: 81%

“…Tables and show convergence properties for WENO5‐Z scheme and the present scheme near the discontinuity, respectively. It should be noted that the error refers to

(||, {\overset{f}{true}}_{i + \frac{1}{2}} - h_{i + \frac{1}{2}})

Section: The New Methodsmentioning

confidence: 99%

Improvement of multistep WENO scheme and its extension to higher orders of accuracy

Yan

Zhu

2016

“…The new scheme has the similar formula as those of the improved multistep WENO scheme suggested by Ma et al 19 ; hence the resulted scheme is also more efficient than the original multistep WENO scheme of Shen et al 16 Meanwhile, theoretical analysis shows that the new scheme can improve the accuracy at transition points without reducing the accuracy in smooth regions including critical points. Then, a high performance fifth-order multistep WENO scheme is proposed by using a nonlinear function of two adjacent local smoothness indicators to replace previous linear combination.…”

Section: Conclusion Remarksmentioning

confidence: 82%

“…Comparing Equation (16) with Equation (18), we can see that the accuracy at transition point is improved by one order. A symmetric scenario, which the discontinuity is present at…”

Section: Accuracy Analysis At Transition Pointmentioning

confidence: 91%

See 1 more Smart Citation

A high performance fifth‐order multistep WENO scheme

Zeng

Shen

Liu

et al. 2019

Self Cite

Summary Many efforts have been made to improve the accuracy of the conventional weighted essentially nonoscillatory (WENO) scheme at transition points (connecting a smooth region and a discontinuity point). This paper analyzes these works and further develops a more effective multistep WENO scheme. Theoretical analysis and numerical results show that the new scheme not only improves the accuracy by one order higher than the traditional fifth‐order WENO schemes at transition point but also maintains the fifth‐order accuracy in smooth regions even at critical point where the first derivative vanishes.

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Peng

Shen

2016

Self Cite

SUMMARYHybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem-dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non-oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high-order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh-order hybrid compact-reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth-order WENO scheme, and it has seventh-order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh-order WENO scheme.