| For numerical simulation of high-speed flows,it requires that discontinuities are stably captured,and small-scale structures are resolved with high-fidelity.These two aspects put forward almost contradictory requirements for commonly used shock-capturing schemes.On the one hand,numerical schemes need to be appropriately dissipative for robust computation and capturing of discontinuity without spurious oscillations;on the other hand,dispersion and dissipation errors of numerical schemes need to be minimized to restore the details of small-scale structures.Existing high-order nonlinear weighted schemes still have some problems remained to be solved,such as numerical oscillation and too much dissipation.In this paper,the spectral characteristics of high-order weighted compact nonlinear schemes(WCNS)are systematically investigated.Three parts of WCNS are optimized to minimize spectral errors,including nonlinear weight function,sick weight in the nonlinear weighting process,and flux difference scheme.First,the classic linear Fourier analysis,approximate dispersion relation,and more advanced nonlinear spectral analysis are applied to investigate spectral characteristics of the third to seventh-order WCNS(including two kinds of hybrid cell-edge and cell-node difference schemes).Nonlinear spectral analysis reveals scattering and overshoot behaviors of the modified wavenumber,such as anti-dissipation.The influence of nonlinearity on the evolution of the energy spectrum of broadband and discontinuous signals is also investigated.Second,based on the smoothness indicators of interpolation sub-stencils,a parameterfree adaptive algorithm is proposed to calculate ε,which is a small quantity in the nonlinear weight functions.The new algorithm not only makes nonlinear weight functions scale-invariant,but also reasonably adjusts the value of ε based on characteristics of local flowfield,which helps to reduce nonlinear error in smooth areas and suppress numerical oscillations near discontinuities.The relation between convergence,critical point and numerical oscillation is also discussed.Then,the sick weight problem in the nonlinear weighting process,such as large nonlinear weights for the purely upwind or downwind sub-stencils in a fifth-order weighted scheme,is newly discovered and optimized.It is shown that sick weight is common in smooth fields,and arises more frequently near a discontinuity.Sick weight can drastically alter the spectral characteristics,such as from dissipative to anti-dissipative,which may lead to large error or instability.A new metric named modified wavenumber component is proposed to investigate the influence of interpolation scheme at individual cell-edge on the final spectral characteristics,which also helps to explicitly explain the overshoot and undershoot behaviors in modified wavenumber for the first time.To alleviate the impacts caused by sick weight,the modified wavenumber components of interpolations on sub-stencils are optimized.Compared with the original scheme,optimized schemes yield greatly suppressed spectral overshoots,improved convergence and less serious post-shock numerical oscillations in steady shock computations.Last but not least,based on the idea of hybrid cell-edge and cell-node difference,new optimized flux difference schemes are proposed.The feasibility of optimizing nonlinear schemes straightforwardly based on approximate dispersion relation is investigated and verified.Compared with the conventional optimization of interpolation scheme,the new optimization of difference scheme shows greater potential,such as lower risk of antidissipation and higher resolving efficiency,without extending stencil or sacrificing formal order.New optimized difference schemes can greatly increase the resolution of wave and vortex structures,especially in long-time simulations. |
