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

Abstract

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.

A high performance fifth‐order multistep WENO scheme

“…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.…”

“…However, limited by the condition of ( k ∕ j , k ≠ j), the simple linear combination of different , such as the formula of Ma et al 19 and van Lith et al, 20 cannot achieve the expected effects. However, limited by the condition of ( k ∕ j , k ≠ j), the simple linear combination of different , such as the formula of Ma et al 19 and van Lith et al, 20 cannot achieve the expected effects.…”

“…16 and HM-WENO achieve fifth-order accuracy at critical point. The IM-WENO scheme 19 has the same fourth-order accuracy as the WENO-Z scheme (q = 1, = 10 −40 ), although its error is less than WENO-Z's. The IM-WENO scheme 19 has the same fourth-order accuracy as the WENO-Z scheme (q = 1, = 10 −40 ), although its error is less than WENO-Z's.…”

“…Meanwhile, similar to the improved multistep WENO scheme proposed by Ma et al, 19 the present method only needs to calculate three nonlinear weights and does not require a mapping function, which makes the algorithm more efficient than the original multistep WENO scheme. Analysis shows that the new method improves the accuracy at transition points, and the calculated weights satisfy the NSCs for the fifth-order convergence in smooth regions even at critical points.…”

“…However, in the multistep WENO scheme, six weights are calculated and the mapping function proposed by Henrick et al 5 is needed, which make the algorithm inefficient. In order to enhance the computational efficiency of the multistep WENO scheme, Ma et al 19 proposed an improved multistep WENO scheme and its extension to higher orders of accuracy. Compared with the multistep WENO scheme, although the improved scheme can greatly improve the computational efficiency, the numerical results show that it cannot effectively reduce the dissipation near the discontinuities and its accuracy at critical point is of fourth order.…”