A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppre...A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics.展开更多
Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer....Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer. For super/hypersonic boundary layers, the external disturbances first interact with the shock ahead of the flying vehicles before entering the boundary layer. Since direct numerical simulation （DNS） is the only available tool for its comprehensive and detailed investigation, an important problem arises whether the numerical scheme, especially the shock-capturing method, can faithfully reproduce the interaction of the external disturbances with the shock, which is so far unknown. This paper is aimed to provide the answer. The interaction of weak disturbances with an oblique shock is investigated, which has a known theoretical solution. Numerical simulation using the shock-capturing method is conducted, and results are compared with those given by theoretical analysis, which shows that the adopted numerical method can faithfully reproduce the interaction of weak external disturbances with the shock.展开更多
Focuses on a study wherein a type of shock fitting method has been used to solve dimensional flow problems with interactions of various discontinuities. Discretization of the transformed equations; Numerical results; ...Focuses on a study wherein a type of shock fitting method has been used to solve dimensional flow problems with interactions of various discontinuities. Discretization of the transformed equations; Numerical results; Interactions of discontinuities.展开更多
Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights,they exhibit distinct numerical dissipation in smooth re...Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights,they exhibit distinct numerical dissipation in smooth regions.This study presents an extension of our previous research which confirmed that adaptively adjusting parameter C can indeed overcome the inadequacy of the usage of a constant small value.Cmin is applied near a discontinuity while Cmax is used elsewhere and they are switched according to the variation of the local flow-field property.This study provides the reference values of the adaptive parameter C of WENOCU4 and systematically evaluates the comprehensive performance of three different switches(labeled as the binary,continuous,and hyperbolic tangent switches,respectively)based on an optimized efficient WENOCU4 scheme(labeled as EWENOCU4).Varieties of 1D scalar equations,empirical dispersion relation analysis,and multi-dimensional benchmark cases of Euler equations are analyzed.Generally,the dissipation and dispersion properties of these three switches are similar.Especially,employing the binary switch,EWENOCU4 achieves the best comprehensive properties.Specifically,the binary switch can efficiently filter more misidentifications in smooth regions than others do,particularly for the cases of 1 D scalar equations and Euler equations.Also,the computational efficiency of the binary switch is superior to that of the hyperbolic tangent switch.Moreover,the optimized scheme exhibits high-resolution spectral properties in the wavenumber space.Therefore,employing the binary switch is a more cost-effective improvement for schemes and is particularly suitable for the simulation of complex shock/turbulence interaction.This study provides useful guidance for the reference values of parameter C and the evaluation of adaptive switches.展开更多
This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we axtend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requir...This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we axtend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the computational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.展开更多
A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into...A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.展开更多
In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adapti...In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adaptive technique avoids the use of redundant meshes in smooth regions,while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning.The compatibility of these two significant features relies on a novel limiter designed for mesh refinements.It ensures the positivity of solutions on all newly created cells.Therefore,the proposed method is completely positivity-preserving and thus highly robust.It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance.The performance of the proposed method is examined in various numerical experiments.When solving Euler equations,the technique of Local Artificial Diffusivity(LAD)is additionally coupled to damp oscillations.More importantly,when solving Navier-Stokes equations,the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly.The implementation of the method becomes rather simple.展开更多
In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena（wave refraction, diffraction, shoaling and bre...In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena（wave refraction, diffraction, shoaling and breaking） is proposed. The model is based on an integral formulation of the Navier-Stokes equations which are solved on a time dependent coordinate system： a coordinate transformation maps the varying coordinates in the physical domain to a uniform transformed space. The equations of motion are discretized by means of a finite-volume shock-capturing numerical procedure based on high order WENO reconstructions. The solution procedure for the equations of motion uses a third order accurate Runge-Kutta（SSPRK） fractional-step method and applies a pressure corrector formulation in order to obtain a divergence-free velocity field at each stage. The proposed model is validated against several benchmark test cases.展开更多
A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the cha...A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes （2nd-order and 3rd-order） and three compact upwind schemes （3rd-order, 5th-order and 7th-order） are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.展开更多
In this paper, a high-resolution, hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique. Firstly, a sufficient condition for a family oftri-dia...In this paper, a high-resolution, hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique. Firstly, a sufficient condition for a family oftri-diagonal compact schemes to have independent dispersion and dissipation is derived. Then, a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed. Finally, the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme. Moreover, the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several test cases are carried out to verify the high- resolution as well as the robust shock-capturing capabilities of the proposed scheme.展开更多
This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38(2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals th...This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38(2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed.With the aid of the direct Eulerian GRP(generalized Riemann problem)methods and the analytical resolution of the local“quasi 1D”GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations.Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.展开更多
The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity appr...The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity approach.The proposed paper describes the numerical methods used in the study,and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems.The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.展开更多
基金The project supported by the National Natural Science Foundation of China (19972038)Foundation of the National CFD Laboratory of China
文摘A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics.
基金supported by the National Natural Science Foundation of China(Nos.11472188 and11332007)the National Key Research and Development Program of China(No.2016YFA0401200)
文摘Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer. For super/hypersonic boundary layers, the external disturbances first interact with the shock ahead of the flying vehicles before entering the boundary layer. Since direct numerical simulation （DNS） is the only available tool for its comprehensive and detailed investigation, an important problem arises whether the numerical scheme, especially the shock-capturing method, can faithfully reproduce the interaction of the external disturbances with the shock, which is so far unknown. This paper is aimed to provide the answer. The interaction of weak disturbances with an oblique shock is investigated, which has a known theoretical solution. Numerical simulation using the shock-capturing method is conducted, and results are compared with those given by theoretical analysis, which shows that the adopted numerical method can faithfully reproduce the interaction of weak external disturbances with the shock.
基金the North Carolina Supercomputing Center, the RGC of Hong Kong,and the FRG of Hong Kong Baptist University.
文摘Focuses on a study wherein a type of shock fitting method has been used to solve dimensional flow problems with interactions of various discontinuities. Discretization of the transformed equations; Numerical results; Interactions of discontinuities.
基金Project supported by the National Natural Science Foundation of China(Nos.11522222,11925207,and 11472305)the Scientific Research Plan of National University of Defense Technology in 2019(No.ZK19-02)the Postgraduate Scientific Research Innovation Project of Hunan Province(Nos.CX20200008 and CX20200084),China。
文摘Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights,they exhibit distinct numerical dissipation in smooth regions.This study presents an extension of our previous research which confirmed that adaptively adjusting parameter C can indeed overcome the inadequacy of the usage of a constant small value.Cmin is applied near a discontinuity while Cmax is used elsewhere and they are switched according to the variation of the local flow-field property.This study provides the reference values of the adaptive parameter C of WENOCU4 and systematically evaluates the comprehensive performance of three different switches(labeled as the binary,continuous,and hyperbolic tangent switches,respectively)based on an optimized efficient WENOCU4 scheme(labeled as EWENOCU4).Varieties of 1D scalar equations,empirical dispersion relation analysis,and multi-dimensional benchmark cases of Euler equations are analyzed.Generally,the dissipation and dispersion properties of these three switches are similar.Especially,employing the binary switch,EWENOCU4 achieves the best comprehensive properties.Specifically,the binary switch can efficiently filter more misidentifications in smooth regions than others do,particularly for the cases of 1 D scalar equations and Euler equations.Also,the computational efficiency of the binary switch is superior to that of the hyperbolic tangent switch.Moreover,the optimized scheme exhibits high-resolution spectral properties in the wavenumber space.Therefore,employing the binary switch is a more cost-effective improvement for schemes and is particularly suitable for the simulation of complex shock/turbulence interaction.This study provides useful guidance for the reference values of parameter C and the evaluation of adaptive switches.
基金This work was supported in part by National Natural Science Foundation of China, the StateMajor Key Project for Basic Research
文摘This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we axtend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the computational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.
基金supported by the State Ocean Administration People’s Republic of China(Grant No.201405025)the Key Laboratory for Sea Area Management Technology(SOA)(Grant No.201603)
文摘A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.
基金supported by the National Natural Science Foundation of China(No.11721202).
文摘In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adaptive technique avoids the use of redundant meshes in smooth regions,while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning.The compatibility of these two significant features relies on a novel limiter designed for mesh refinements.It ensures the positivity of solutions on all newly created cells.Therefore,the proposed method is completely positivity-preserving and thus highly robust.It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance.The performance of the proposed method is examined in various numerical experiments.When solving Euler equations,the technique of Local Artificial Diffusivity(LAD)is additionally coupled to damp oscillations.More importantly,when solving Navier-Stokes equations,the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly.The implementation of the method becomes rather simple.
文摘In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena（wave refraction, diffraction, shoaling and breaking） is proposed. The model is based on an integral formulation of the Navier-Stokes equations which are solved on a time dependent coordinate system： a coordinate transformation maps the varying coordinates in the physical domain to a uniform transformed space. The equations of motion are discretized by means of a finite-volume shock-capturing numerical procedure based on high order WENO reconstructions. The solution procedure for the equations of motion uses a third order accurate Runge-Kutta（SSPRK） fractional-step method and applies a pressure corrector formulation in order to obtain a divergence-free velocity field at each stage. The proposed model is validated against several benchmark test cases.
基金Project supported by the National Natural Science Foundation of China (Nos.10321002 and 10672012)
文摘A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes （2nd-order and 3rd-order） and three compact upwind schemes （3rd-order, 5th-order and 7th-order） are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.
基金supported by the National Natural Science Foundation of China(Grant No.11302250)a National University Research Grant of Xi’an Research Institute of High-tech(Grant No.2013QNJJ029)
文摘In this paper, a high-resolution, hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique. Firstly, a sufficient condition for a family oftri-diagonal compact schemes to have independent dispersion and dissipation is derived. Then, a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed. Finally, the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme. Moreover, the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several test cases are carried out to verify the high- resolution as well as the robust shock-capturing capabilities of the proposed scheme.
基金The authors were partially supported by the Special Project on High-performance Computing under the National Key R&D Program(No.2016YF B0200603)Sci-ence Challenge Project(No.JCK Y2016212A502)the National Natural Science Foundation of China(Nos.91630310&11421101).
文摘This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38(2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed.With the aid of the direct Eulerian GRP(generalized Riemann problem)methods and the analytical resolution of the local“quasi 1D”GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations.Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.
基金supported by the Russian Science Foundation(project 22-11-00199).The author would like to thank Tatiana Zezyulina for the qualified assistance in English and the reviewers for their helpful comments.
文摘The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity approach.The proposed paper describes the numerical methods used in the study,and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems.The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.
