Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws.The family of minmod limiters serves as the prototype example.Here,we revisi...Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws.The family of minmod limiters serves as the prototype example.Here,we revisit the question of non-oscillatory behavior of high-resolution central schemes in terms of the slope limiter proposed by van Albada et al.(Astron Astrophys 108:76–84,1982).The van Albada(vA)limiter is smoother near extrema,and consequently,in many cases,it outperforms the results obtained using the standard minmod limiter.In particular,we prove that the vA limiter ensures the one-dimensional Total-Variation Diminishing(TVD)stability and demonstrate that it yields noticeable improvement in computation of one-and two-dimensional systems.展开更多
By conjugating features of combustion gas jetting flows of the solid-rocket and using mathematical methods, a numerical scheme is systematically derived based on Harten′s standard TVD scheme, which fits for the flow ...By conjugating features of combustion gas jetting flows of the solid-rocket and using mathematical methods, a numerical scheme is systematically derived based on Harten′s standard TVD scheme, which fits for the flow with high temperature, pressure and velocity. The rational calculation formula of pressure partial derivation is also given out. By using the chemical kinetics knowledge, problems of multi-component and finite rate chemical reaction contained in combustion gas of the rocket flow field are discussed. The method for solving the mass source term of chemical reaction is clarified. Taking 9 reaction equations with 12 components as an example and utilizing the established calculation program, the free jetting flow field of the rocket is simulated. Numerical results show the correctness of the numerical scheme.展开更多
A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can e...A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.展开更多
In this paper, a new numerical scheme of Total Variation Diminishing (TVD) MacCormack type for MagnetoHydroDynamic (MHD) equations is proposed by taking into account of the characteristics such as convergence, stabili...In this paper, a new numerical scheme of Total Variation Diminishing (TVD) MacCormack type for MagnetoHydroDynamic (MHD) equations is proposed by taking into account of the characteristics such as convergence, stability, resolution. This new scheme is established by solving the MHD equations with a TVD modified MacCormack scheme for the purpose of developing a scheme of quick convergence as well as of TVD property. To show the validation, simplicity and practicability of the scheme for modelling MHD problems, a self-similar Cauchy problem with the discontinuous initial data consisting of constant states, and the collision of two fast MHD shocks, and two-dimensional Orszag and Tang's MHD vortex problem are discussed with the numerical results conforming to the existing results obtained by the Roe type TVD, the high-order Godunov scheme,and Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical tests show that this two-step TVD MacCormack numerical scheme for MHD system is of robust operation in the presence of very strong waves, thin shock fronts, thin contact and slip surface discontinuities.展开更多
In order to investigate parameters of FAE (fuel air explosive) explosion, the two-phase gas-droplet conservation equations with two-dimensional axial symmetry in the Euler coordinate were used. High-resolution implici...In order to investigate parameters of FAE (fuel air explosive) explosion, the two-phase gas-droplet conservation equations with two-dimensional axial symmetry in the Euler coordinate were used. High-resolution implicit TVD ( total variation diminishing) schemes were applied to gas phase equations and MacCormack schemes to liquid equations. The formation and propagation of gas-droplet detonation wave were simulated numerically. The simulation results and the others are compared with a good agreement.展开更多
Since governing equations are discretized using a finite volume method for FV TVD scheme, we use integral governing equations to solve the flow field. We achieve N S equations in terms of cylinder coordinate velocity ...Since governing equations are discretized using a finite volume method for FV TVD scheme, we use integral governing equations to solve the flow field. We achieve N S equations in terms of cylinder coordinate velocity components in an arbitrary curvilinear coordinate using a tensor analytic method to the integral governing equations. It’s also testified that term g which include Jacobian can be reduced when governing equations are discretized on an infinitesimal small control volume. Numerical calculations indicated that this scheme can capture shocks and contact discontinuities exactly and the solution with this treatment is in good agreement with the experimental data.展开更多
In most TVD schemes, the r-factors were proposed according to the cell-centered（CC） finite volume method（FVM） framework for the numerical approximation to the convective term. However, it is questionable whether t...In most TVD schemes, the r-factors were proposed according to the cell-centered（CC） finite volume method（FVM） framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered（VC） FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner（1996）, Darwish and Moukalled（2003） and Cassuli and Zanolli（2005） can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely Фu in the paper, was validated by the tests to effectively dissipate spurious oscillations.展开更多
A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing e...A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.展开更多
基金Research was supported in part by the ONR Grant N00014-2112773.
文摘Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws.The family of minmod limiters serves as the prototype example.Here,we revisit the question of non-oscillatory behavior of high-resolution central schemes in terms of the slope limiter proposed by van Albada et al.(Astron Astrophys 108:76–84,1982).The van Albada(vA)limiter is smoother near extrema,and consequently,in many cases,it outperforms the results obtained using the standard minmod limiter.In particular,we prove that the vA limiter ensures the one-dimensional Total-Variation Diminishing(TVD)stability and demonstrate that it yields noticeable improvement in computation of one-and two-dimensional systems.
文摘By conjugating features of combustion gas jetting flows of the solid-rocket and using mathematical methods, a numerical scheme is systematically derived based on Harten′s standard TVD scheme, which fits for the flow with high temperature, pressure and velocity. The rational calculation formula of pressure partial derivation is also given out. By using the chemical kinetics knowledge, problems of multi-component and finite rate chemical reaction contained in combustion gas of the rocket flow field are discussed. The method for solving the mass source term of chemical reaction is clarified. Taking 9 reaction equations with 12 components as an example and utilizing the established calculation program, the free jetting flow field of the rocket is simulated. Numerical results show the correctness of the numerical scheme.
基金Project supported by the National Natural Science Foundation of China (Nos. 10172015 and 90205010)
文摘A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.
基金the National Natural Science Foundation of China(No.49925412,49990450),the National Basic Research Science Foundation(No.G2000078405)
文摘In this paper, a new numerical scheme of Total Variation Diminishing (TVD) MacCormack type for MagnetoHydroDynamic (MHD) equations is proposed by taking into account of the characteristics such as convergence, stability, resolution. This new scheme is established by solving the MHD equations with a TVD modified MacCormack scheme for the purpose of developing a scheme of quick convergence as well as of TVD property. To show the validation, simplicity and practicability of the scheme for modelling MHD problems, a self-similar Cauchy problem with the discontinuous initial data consisting of constant states, and the collision of two fast MHD shocks, and two-dimensional Orszag and Tang's MHD vortex problem are discussed with the numerical results conforming to the existing results obtained by the Roe type TVD, the high-order Godunov scheme,and Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical tests show that this two-step TVD MacCormack numerical scheme for MHD system is of robust operation in the presence of very strong waves, thin shock fronts, thin contact and slip surface discontinuities.
文摘In order to investigate parameters of FAE (fuel air explosive) explosion, the two-phase gas-droplet conservation equations with two-dimensional axial symmetry in the Euler coordinate were used. High-resolution implicit TVD ( total variation diminishing) schemes were applied to gas phase equations and MacCormack schemes to liquid equations. The formation and propagation of gas-droplet detonation wave were simulated numerically. The simulation results and the others are compared with a good agreement.
文摘Since governing equations are discretized using a finite volume method for FV TVD scheme, we use integral governing equations to solve the flow field. We achieve N S equations in terms of cylinder coordinate velocity components in an arbitrary curvilinear coordinate using a tensor analytic method to the integral governing equations. It’s also testified that term g which include Jacobian can be reduced when governing equations are discretized on an infinitesimal small control volume. Numerical calculations indicated that this scheme can capture shocks and contact discontinuities exactly and the solution with this treatment is in good agreement with the experimental data.
基金financially supported by the National Natural Science Foundation of China(Grant Nos.41306078 and 41301414)the National Engineering Research Center for Inland Waterway Regulation and Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education Program(Grant No.SLK2016B03)the Key Laboratory of the Inland Waterway Regulation of the Ministry of Transportation Program(Grant No.NHHD-201514)
文摘In most TVD schemes, the r-factors were proposed according to the cell-centered（CC） finite volume method（FVM） framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered（VC） FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner（1996）, Darwish and Moukalled（2003） and Cassuli and Zanolli（2005） can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely Фu in the paper, was validated by the tests to effectively dissipate spurious oscillations.
基金supported by the National Natural Science Foundation of China(Grant No.51579034)the Open Fund of the Key Laboratory of Ocean Circulation and Waves,Chinese Academy of Sciences(Grant No.KLOCW1502)
文摘A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.
