In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an...In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function(ENO-property),a scaleinvariant property with an arbitrary scale of a function(Si-property),and an optimal order of accuracy with smooth function regardless of the critical point(Cp-property).The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically,which is caused by a loss of sub-stencils’adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor.A new nonlinear weight is devised by using an average of the function values and the descaling function,providing the new WCNS schemes(WCNS-Zm/Dm)with many attractive properties.The ENO-property,Si-property and Cp-property of the new WCNS schemes are validated numerically.Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property,while only the WCNS-Dm scheme satisfies the Cp-property.In addition,the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved.Several one-dimensional shock tube problems,and two-dimensional double Mach reflection(DMR)problem and the Riemann IVP problem are simulated to illustrate the ENOproperty and Si-property of the scale-invariant WCNS-Zm/Dm schemes.展开更多
In this paper,we present a hybrid form of weighted compact nonlinear scheme(WCNS)for hyperbolic conservation laws by applying linear and nonlinear methods for smooth and discontinuous zones individually.To fulfill thi...In this paper,we present a hybrid form of weighted compact nonlinear scheme(WCNS)for hyperbolic conservation laws by applying linear and nonlinear methods for smooth and discontinuous zones individually.To fulfill this algorithm,it is inseparable from the recognition ability of the discontinuity detector adopted.In specific,a troubled-cell indicator is utilized to recognize unsmooth areas such as shock waves and contact discontinuities,while avoiding misjudgments of smooth structures.Some classical detectors are classified into three basic types:derivative combination,smoothness indicators and characteristic decomposition.Meanwhile,a new improved detector is proposed for comparison.Then they are analyzed through identifying a series of waveforms firstly.After that,hybrid schemes using such indicators,as well as different detection variables,are examined with Euler equations,so as to investigate their ability to distinguish practical discontinuities on various levels.Simulation results demonstrate that the proposed algorithm has similar performances to pure WCNS,while it generally saves 50 percent of CPU time for 1D cases and about 40 percent for 2D Euler equations.Current research is in the hope of providing some reference and establishing some standards for judging existing discontinuity detectors and developing novel ones.展开更多
In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws...In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.展开更多
To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal...To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal orders,or extending stencil widths,are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes.Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR)for nonlinear schemes.Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators.The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes,in terms of accuracy and resolution for smooth wave and vortex,especially for long-time simulations.Using optimized schemes increases the total CPU time by less than 4%.展开更多
基金supported by the Hunan Provincial Natural Science Foundation of China(No.2022JJ40539)National Natural Science Foundation of China(No.11972370)National Key Project(No.GJXM92579).
文摘In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function(ENO-property),a scaleinvariant property with an arbitrary scale of a function(Si-property),and an optimal order of accuracy with smooth function regardless of the critical point(Cp-property).The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically,which is caused by a loss of sub-stencils’adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor.A new nonlinear weight is devised by using an average of the function values and the descaling function,providing the new WCNS schemes(WCNS-Zm/Dm)with many attractive properties.The ENO-property,Si-property and Cp-property of the new WCNS schemes are validated numerically.Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property,while only the WCNS-Dm scheme satisfies the Cp-property.In addition,the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved.Several one-dimensional shock tube problems,and two-dimensional double Mach reflection(DMR)problem and the Riemann IVP problem are simulated to illustrate the ENOproperty and Si-property of the scale-invariant WCNS-Zm/Dm schemes.
基金supported by the National Natural Science Foundation of China(grant Nos.11972370,92252101,11927803).
文摘In this paper,we present a hybrid form of weighted compact nonlinear scheme(WCNS)for hyperbolic conservation laws by applying linear and nonlinear methods for smooth and discontinuous zones individually.To fulfill this algorithm,it is inseparable from the recognition ability of the discontinuity detector adopted.In specific,a troubled-cell indicator is utilized to recognize unsmooth areas such as shock waves and contact discontinuities,while avoiding misjudgments of smooth structures.Some classical detectors are classified into three basic types:derivative combination,smoothness indicators and characteristic decomposition.Meanwhile,a new improved detector is proposed for comparison.Then they are analyzed through identifying a series of waveforms firstly.After that,hybrid schemes using such indicators,as well as different detection variables,are examined with Euler equations,so as to investigate their ability to distinguish practical discontinuities on various levels.Simulation results demonstrate that the proposed algorithm has similar performances to pure WCNS,while it generally saves 50 percent of CPU time for 1D cases and about 40 percent for 2D Euler equations.Current research is in the hope of providing some reference and establishing some standards for judging existing discontinuity detectors and developing novel ones.
基金Project supported by the National Natural Science Foundation of China(No.11571366)the Basic Research Foundation of National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)
文摘In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.
基金Project supported by the National Key Project(No.GJXM92579)the Defense Industrial Technology Development Program(No.C1520110002)the State Administration of Science,Technology and Industry for National Defence,China。
文摘To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal orders,or extending stencil widths,are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes.Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR)for nonlinear schemes.Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators.The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes,in terms of accuracy and resolution for smooth wave and vortex,especially for long-time simulations.Using optimized schemes increases the total CPU time by less than 4%.
