New nonlinear weights for improving accuracy and resolution of weighted compact nonlinear scheme

Abstract

This paper proposes a new kind of nonlinear weights to improve accuracy and resolution of high-order weighted compact nonlinear scheme. The new nonlinear weights are constructed based on not only the ratios between different smoothness indicators but also their values. The values of smoothness indicators are explicitly considered in the basic formulas of the new nonlinear weights after a careful analysis of convergence accuracy and shock capturing property. The new nonlinear weights approach to the optimal weights as the smoothness indicators approaching zero. Thus, the new nonlinear weights are close to the optimal weights in smooth regions where the smoothness indicators are small. Therefore, optimal order accuracy is maintained in smooth regions. In addition, near discontinuities the new nonlinear weights degenerate to the original nonlinear weights used for designing. Thus, discontinuity capturing ability is ensured. Numerical results show that the weighted compact nonlinear scheme with the new nonlinear weights achieves optimal order accuracy even near high-order critical points, captures discontinuities sharply without obvious oscillation, has higher resolution and higher efficiency than other nonlinear schemes and has obvious advantage in capturing small scale structures.