Based on the traditional fifth-order weighted essentially non-oscillatory(WENO)scheme,a smoothness indicator is introduced to improve the capability of WENO schemes for resolving short waves.In the construction of the...Based on the traditional fifth-order weighted essentially non-oscillatory(WENO)scheme,a smoothness indicator is introduced to improve the capability of WENO schemes for resolving short waves.In the construction of the new smoothness indicator,the proportion of the first-order term in the original smoothness indicator is reduced by replacing the square of the first-order term with the product of the first-order and the third-order terms.To preserve the fifth-order of convergence rate,the smoothness indicator is combined with the method of Borges,et al.The numerical results show that the proposed schemes are more suitable for simulating turbulent flows or aeroacoustics problems than the previous fifth-order WENO schemes,thanks to its improved resolution on short waves.展开更多
In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated...In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.展开更多
In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridizat...In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridization,the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region;meanwhile,the essentially oscillation-free solution could also be obtained.By adapting the original smoothness indicator in the WENO reconstruction,the stencil is distinguished into three types:smooth,non-smooth,and high-frequency region.In the smooth region,the linear reconstruction is used and the non-smooth region with the WENO reconstruction.In the high-frequency region,the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor,which could capture high-frequency wave efficiently.Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.展开更多
In this article,novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi eq...In this article,novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations.These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil.The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution.Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.展开更多
Recently,the China Textile Industry Association held 2011 textile and economic operation analysis meeting.The meeting held that the 2011 overall economic operation of China’s textile industry was stable,the indicator...Recently,the China Textile Industry Association held 2011 textile and economic operation analysis meeting.The meeting held that the 2011 overall economic operation of China’s textile industry was stable,the indicators are normal,but various types of risk faced by the operation was展开更多
Weighted essentially non-oscillatory(WENO)schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and ...Weighted essentially non-oscillatory(WENO)schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions.However,like many other high-order shock capturing schemes,WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave.This is a long-standing difficulty for high-order shock capturing schemes.In recent years,this non-convergence problem has been studied extensively for WENO schemes.Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations,which are at the small local truncation error level but prevent the residue to settle down to machine zero.Several strategies have been proposed to reduce these slight post shock oscillations,including the design of new smoothness indicators for the fifth-order WENO scheme,the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy,and the design of a new type of WENO schemes.With these strategies,the convergence to steady states is improved significantly.Moreover,the strategies are applicable to other types of weighted schemes.In this paper,we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.展开更多
This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise we...This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory(WENO)finite difference schemes.Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems,and the twodimensional Rayleigh-Taylor instability(RTI)problem as examples,we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition,which are often given and used in the literature,and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem.An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error.The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability,reduced rounding errors and increased computational efficiency.展开更多
In this paper,a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework.The new scheme is a convex combination of two quadratic polynomials w...In this paper,a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework.The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion.The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained.The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy.The linear advection equation,nonlinear scalar Burgers equation,and one-and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method.Finally,one-and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.展开更多
基金Supported by the National Natural Science Foundation of China(50830201,11102179)the Nanjing University of Aeronautics and Astronautics Research Funding(NP 2011033)
文摘Based on the traditional fifth-order weighted essentially non-oscillatory(WENO)scheme,a smoothness indicator is introduced to improve the capability of WENO schemes for resolving short waves.In the construction of the new smoothness indicator,the proportion of the first-order term in the original smoothness indicator is reduced by replacing the square of the first-order term with the product of the first-order and the third-order terms.To preserve the fifth-order of convergence rate,the smoothness indicator is combined with the method of Borges,et al.The numerical results show that the proposed schemes are more suitable for simulating turbulent flows or aeroacoustics problems than the previous fifth-order WENO schemes,thanks to its improved resolution on short waves.
基金the National Natural Science Foundation of China(91641107,91852116,12071470)Fundamental Research of Civil Aircraft(MJ-F-2012-04)of Ministry of Industrialization and Information of China.
文摘In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.
基金the National Numerical Windtunnel Project NNW2019ZT4-B08the NSFC grant No.11871449.
文摘In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridization,the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region;meanwhile,the essentially oscillation-free solution could also be obtained.By adapting the original smoothness indicator in the WENO reconstruction,the stencil is distinguished into three types:smooth,non-smooth,and high-frequency region.In the smooth region,the linear reconstruction is used and the non-smooth region with the WENO reconstruction.In the high-frequency region,the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor,which could capture high-frequency wave efficiently.Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.
文摘In this article,novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations.These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil.The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution.Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.
文摘Recently,the China Textile Industry Association held 2011 textile and economic operation analysis meeting.The meeting held that the 2011 overall economic operation of China’s textile industry was stable,the indicators are normal,but various types of risk faced by the operation was
基金The work of the first author was supported by NSFC grant 11732016The research of the second author was supported by NSFC grant 11872210The research of the third author was supported by NSF grant DMS-1719410.
文摘Weighted essentially non-oscillatory(WENO)schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions.However,like many other high-order shock capturing schemes,WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave.This is a long-standing difficulty for high-order shock capturing schemes.In recent years,this non-convergence problem has been studied extensively for WENO schemes.Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations,which are at the small local truncation error level but prevent the residue to settle down to machine zero.Several strategies have been proposed to reduce these slight post shock oscillations,including the design of new smoothness indicators for the fifth-order WENO scheme,the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy,and the design of a new type of WENO schemes.With these strategies,the convergence to steady states is improved significantly.Moreover,the strategies are applicable to other types of weighted schemes.In this paper,we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.
基金The authors would like to acknowledge the funding support of this research by the National Natural Science Foundation of China(Nos.11801383,11871443)National Science and Technology Major Project(No.20101010)+2 种基金Shandong Provincial Natural Science Foundation(No.ZR2017MA016)Fundamental Research Funds for the Central Universities(No.201562012)The authors(Li and Don)also like to thank Shijiazhuang Tiedao University and Ocean University of China for providing the startup funds(No.Z6811021064 and 201712011),respectively.
文摘This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory(WENO)finite difference schemes.Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems,and the twodimensional Rayleigh-Taylor instability(RTI)problem as examples,we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition,which are often given and used in the literature,and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem.An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error.The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability,reduced rounding errors and increased computational efficiency.
基金supported by National Natural Science Foundation of China(grant No.51579206)the Key R&D Projects in Shaanxi Province,China(grant No.2018SF-352)the Water Conservancy Science and Technology Project of Shaanxi Province,China(grant No.2017slkj-17).
文摘In this paper,a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework.The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion.The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained.The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy.The linear advection equation,nonlinear scalar Burgers equation,and one-and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method.Finally,one-and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.
