For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carrie...For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.展开更多
Direct simulation of 3-D MHD(magnetohydrodynamics)flows in liquid metal fusion blanket with flow channel insert(FCI)has been conducted.Two kinds of pressure equilibrium slot(PES)in FCI,which are used to balance the pr...Direct simulation of 3-D MHD(magnetohydrodynamics)flows in liquid metal fusion blanket with flow channel insert(FCI)has been conducted.Two kinds of pressure equilibrium slot(PES)in FCI,which are used to balance the pressure difference between the inside and outside of FCI,are considered with a slot in Hartmann wall or a slot in side wall,respectively.The velocity and pressure distribution of FCI made of SiC/SiC_f are numerically studied to illustrate the 3-D MHD flow effects,which clearly show that the flows in fusion blanket with FCI are typical three-dimensional issues and the assumption of 2-D fully developed flows is not the real physical problem of the MHD flows in dual-coolant liquid metal fusion blanket.The optimum opening location of PES has been analyzed based on the 3-D pressure and velocity distributions.展开更多
The work presents new methods for selecting adaptive artificial viscosity(AAV)in iterative algorithms of completely conservative difference schemes(CCDS)used to solve gas dynamics equations in Euler variables.These me...The work presents new methods for selecting adaptive artificial viscosity(AAV)in iterative algorithms of completely conservative difference schemes(CCDS)used to solve gas dynamics equations in Euler variables.These methods allow to effectively suppress oscillations,including in velocity profiles,as well as computational instabilities in modeling gas-dynamic processes described by hyperbolic equations.The methods can be applied both in explicit and implicit(method of separate sweeps)iterative processes in numerical modeling of gas dynamics in the presence of heat and mass transfer,as well as in solving problems of magnetohydrodynamics and computational astrophysics.In order to avoid loss of solution accuracy on spatially non-uniform grids,in this work an algorithm of grid embeddings is developed,which is applied near transition points between cells of different sizes.The developed algorithms of CCDS using the methods for AAV selection and the algorithm of grid embeddings are implemented for various iterative processes.Calculations are performed for the classical problem of decay of an arbitrary discontinuity(Sod’s problem)and the problem of propagation of two symmetric rarefaction waves in opposite directions(Einfeldt’s problem).In the case of using different methods for selecting the AAV,a comparison of the solutions of the Sod’s problem on uniform and non-uniform grids and a comparison of the solutions of the Einfeldt’s problem on a uniform grid are performed.As a result of the comparative analysis,the applicability of these methods is shown in the spatially one-dimensional case(explicit and implicit iterative processes).The obtained results are compared with the data from the literature.The results coincide with analytical solutions with high accuracy,where the relative error does not exceed 0.1%,which demonstrates the effectiveness of the developed algorithms and methods.展开更多
In this paper,we develop a fourth-order conservative wavelet-based shock-capturing scheme.The scheme is constructed by combining a wavelet collocation upwind method with the monotonic tangent of hyperbola for interfac...In this paper,we develop a fourth-order conservative wavelet-based shock-capturing scheme.The scheme is constructed by combining a wavelet collocation upwind method with the monotonic tangent of hyperbola for interface capturing(THINC)technique.We employ boundary variation diminishing(BVD)reconstruction to enhance the scheme’s effectiveness in handling shocks.First,we prove that wavelet collocation upwind schemes based on interpolating wavelets can be reformulated into a conservative form within the framework of wavelet theory,forming the foundation of the proposed scheme.The new fourthorder accurate scheme possesses significantly better spectral resolution than the fifth-and even seventh-order WENO-Z(weighted essentially non-oscillatory)schemes over the entire wave-number range.Moreover,the inherent low-pass filtering property of the wavelet bases allows them to filter high-frequency numerical oscillations,endowing the wavelet upwind scheme with robustness and accuracy in solving problems under extreme conditions.Notably,due to the wavelet multiresolution approximation,the proposed scheme possesses a distinctive shape-preserving property absent in the WENO-Z schemes and the fifth-order schemes with BVD reconstruction based on polynomials.Furthermore,compared to the fifth-order scheme with BVD reconstruction based on polynomials—which is significantly superior to the WENO schemes—the proposed scheme further enhances the ability to capture discontinuities.展开更多
The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the ...The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.展开更多
Based on the forced dissipetive nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipotive nonlinear atmos...Based on the forced dissipetive nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipotive nonlinear atmospheric and oceanic equations is analyzed and the computationally stable explicit complete square conservative difference schemes are constructed. The theoretical analysis and numerical experiment prove that the explicit complete square conservative difference schemes are computationally stable and deserve to be disseminated.展开更多
In order to improve the accuracy of forecasts of atmospheric and oceanic phenomena which possess a wide range of space and time scales, it is crucial to design the high-order and stable schemes. On the basis of the ex...In order to improve the accuracy of forecasts of atmospheric and oceanic phenomena which possess a wide range of space and time scales, it is crucial to design the high-order and stable schemes. On the basis of the explicit square-conservative scheme, a high-order compact explicit square-conservative scheme is proposed in this paper. This scheme not only keeps the square-conservative characteristics, but also is of high accuracy. The numerical example shows that this scheme has less computing errors and better computational stability, and it could be considered to be tested and used in many atmospheric and oceanic problems.展开更多
A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CW...A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.展开更多
The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and...The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.展开更多
A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equation...A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.展开更多
A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of ...A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.展开更多
A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries （KdV） equation. A nume...A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries （KdV） equation. A numerical simulation is conducted by the first-order accurate Lax-Friedrichs scheme, which is known for its ability to capture the entropy solution to hyperbolic conservation laws. Periodic boundary conditions are applied to simulate a lengthy propagation, where the profile of the derived KdV solution is taken as the initial condition to observe the change of the profile. The simulation shows good agreement between the approximated KdV solution and the numerical solution.展开更多
Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on cur...Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics.The main purpose of this work is to demonstrate our claim in[3]that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges,which restricts the accuracy of the resulting scheme to at most second order.The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes.Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.展开更多
Based on an adaptive mesh refinement(AMR)technique,a multi-fluid MHD(magnetohydrodynamics)flow is directly simulated by using volume of fluid(VOF)method to capture the interface and charge conservation(consistent and ...Based on an adaptive mesh refinement(AMR)technique,a multi-fluid MHD(magnetohydrodynamics)flow is directly simulated by using volume of fluid(VOF)method to capture the interface and charge conservation(consistent and conservative)scheme to calculate the Lorentz force.Effects of magnetic field on bubble rising in a liquid are conducted,in which the magnetic strength and magnetic directions decide the trajectory and the velocity of rising bubble. The history of kinetic distribution and trajectory is illustrated to describe the effect of the magnetic field on bubble driven flows.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 conservative fifth-order upwind compact scheme using centered stencil is introduced.This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric.Theoretical an...In this paper,a conservative fifth-order upwind compact scheme using centered stencil is introduced.This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric.Theoretical analysis shows that the proposed scheme is low-dissipative and has a relatively large stability range.To maintain the convergence rate of the whole spatial discretization,a proper non-periodic boundary scheme is also proposed.A detailed analysis shows that the spatial discretization implemented with the boundary scheme proposed by Pirozzoli[J.Comput.Phys.,178(2001),pp.81–117]is approximately fourth-order.Furthermore,a hybridmethodology,coupling the compact scheme with WENO scheme,is adopted for problems with discontinuities.Numerical results demonstrate the effectiveness of the proposed scheme.展开更多
In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we...In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.展开更多
In accordance with a new compensation principle of discrete computations,the traditional meteo- rological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy cons...In accordance with a new compensation principle of discrete computations,the traditional meteo- rological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy conservative fidelity schemes,thus resolving the problems of both nonlinear computa- tional instability and incomplete energy conservation,and raising the computational efficiency of the traditional schemes. As the numerical tests of the new schemes demonstrate,in solving the problem of energy conser- vation in operational computations,the new schemes can eliminate the (nonlinear) computational in- stability and,to some extent even the (nonlinear) computational diverging as found in the traditional schemes,Further contrasts between new and traditional schemes also indicate that,in discrete opera- tional computations,the new scheme in the case of nondivergence is capable of prolonging the valid in- tegral time of the corresponding traditional scheme,and eliminating certain kind of systematical com- putational“climate drift”,meanwhile increasing its computational accuracy and reducing its amount of computation.The working principle of this paper is also applicable to the problem concerning baroclin- ic primitive equations.展开更多
In this paper, a numerical method is given to solve relativistic hydrodynamic equations with source terms by conservative finite difference scheme. In calculation, QGP (quark gluon plasma) phase transition is als...In this paper, a numerical method is given to solve relativistic hydrodynamic equations with source terms by conservative finite difference scheme. In calculation, QGP (quark gluon plasma) phase transition is also considered. The numerical experiments have verified the effectiveness of the numerical method and some computational results are illustrated.展开更多
A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is pr...A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.展开更多
文摘For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.
基金supported by National Natural Science Foundation of China with grant Nos.10872212,50936006National Magnetic Confinement Fusion Science Program in China with grant No.2009GB10401
文摘Direct simulation of 3-D MHD(magnetohydrodynamics)flows in liquid metal fusion blanket with flow channel insert(FCI)has been conducted.Two kinds of pressure equilibrium slot(PES)in FCI,which are used to balance the pressure difference between the inside and outside of FCI,are considered with a slot in Hartmann wall or a slot in side wall,respectively.The velocity and pressure distribution of FCI made of SiC/SiC_f are numerically studied to illustrate the 3-D MHD flow effects,which clearly show that the flows in fusion blanket with FCI are typical three-dimensional issues and the assumption of 2-D fully developed flows is not the real physical problem of the MHD flows in dual-coolant liquid metal fusion blanket.The optimum opening location of PES has been analyzed based on the 3-D pressure and velocity distributions.
基金carried out within the framework of the state assignment of KIAM RAS(No.125020701776-0).
文摘The work presents new methods for selecting adaptive artificial viscosity(AAV)in iterative algorithms of completely conservative difference schemes(CCDS)used to solve gas dynamics equations in Euler variables.These methods allow to effectively suppress oscillations,including in velocity profiles,as well as computational instabilities in modeling gas-dynamic processes described by hyperbolic equations.The methods can be applied both in explicit and implicit(method of separate sweeps)iterative processes in numerical modeling of gas dynamics in the presence of heat and mass transfer,as well as in solving problems of magnetohydrodynamics and computational astrophysics.In order to avoid loss of solution accuracy on spatially non-uniform grids,in this work an algorithm of grid embeddings is developed,which is applied near transition points between cells of different sizes.The developed algorithms of CCDS using the methods for AAV selection and the algorithm of grid embeddings are implemented for various iterative processes.Calculations are performed for the classical problem of decay of an arbitrary discontinuity(Sod’s problem)and the problem of propagation of two symmetric rarefaction waves in opposite directions(Einfeldt’s problem).In the case of using different methods for selecting the AAV,a comparison of the solutions of the Sod’s problem on uniform and non-uniform grids and a comparison of the solutions of the Einfeldt’s problem on a uniform grid are performed.As a result of the comparative analysis,the applicability of these methods is shown in the spatially one-dimensional case(explicit and implicit iterative processes).The obtained results are compared with the data from the literature.The results coincide with analytical solutions with high accuracy,where the relative error does not exceed 0.1%,which demonstrates the effectiveness of the developed algorithms and methods.
基金supported by the National Natural Science Foundation of China(Grant No.11925204).
文摘In this paper,we develop a fourth-order conservative wavelet-based shock-capturing scheme.The scheme is constructed by combining a wavelet collocation upwind method with the monotonic tangent of hyperbola for interface capturing(THINC)technique.We employ boundary variation diminishing(BVD)reconstruction to enhance the scheme’s effectiveness in handling shocks.First,we prove that wavelet collocation upwind schemes based on interpolating wavelets can be reformulated into a conservative form within the framework of wavelet theory,forming the foundation of the proposed scheme.The new fourthorder accurate scheme possesses significantly better spectral resolution than the fifth-and even seventh-order WENO-Z(weighted essentially non-oscillatory)schemes over the entire wave-number range.Moreover,the inherent low-pass filtering property of the wavelet bases allows them to filter high-frequency numerical oscillations,endowing the wavelet upwind scheme with robustness and accuracy in solving problems under extreme conditions.Notably,due to the wavelet multiresolution approximation,the proposed scheme possesses a distinctive shape-preserving property absent in the WENO-Z schemes and the fifth-order schemes with BVD reconstruction based on polynomials.Furthermore,compared to the fifth-order scheme with BVD reconstruction based on polynomials—which is significantly superior to the WENO schemes—the proposed scheme further enhances the ability to capture discontinuities.
文摘The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.
基金the Outstanding State Key Laboratory Project of National Science Foundation of China (Grant No. 40023001 )the Key Innovatio
文摘Based on the forced dissipetive nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipotive nonlinear atmospheric and oceanic equations is analyzed and the computationally stable explicit complete square conservative difference schemes are constructed. The theoretical analysis and numerical experiment prove that the explicit complete square conservative difference schemes are computationally stable and deserve to be disseminated.
文摘In order to improve the accuracy of forecasts of atmospheric and oceanic phenomena which possess a wide range of space and time scales, it is crucial to design the high-order and stable schemes. On the basis of the explicit square-conservative scheme, a high-order compact explicit square-conservative scheme is proposed in this paper. This scheme not only keeps the square-conservative characteristics, but also is of high accuracy. The numerical example shows that this scheme has less computing errors and better computational stability, and it could be considered to be tested and used in many atmospheric and oceanic problems.
基金the National Natural Science Foundation of China (60134010)The English text was polished by Yunming Chen.
文摘A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.
文摘The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.
文摘A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.
基金The Project Supported by National Natural Science Foundation of China.
文摘A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.
基金Project supported by the National Natural Science Foundation of China (Nos. 11072141 and 11272199)the National Basic Research Program of China (No. 2012CB725404)+1 种基金the University Research Committee, HKU SPACE Research FundFaculty of Engineering Top-up Grant of the University of Hong Kong (No. 201007176059)
文摘A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries （KdV） equation. A numerical simulation is conducted by the first-order accurate Lax-Friedrichs scheme, which is known for its ability to capture the entropy solution to hyperbolic conservation laws. Periodic boundary conditions are applied to simulate a lengthy propagation, where the profile of the derived KdV solution is taken as the initial condition to observe the change of the profile. The simulation shows good agreement between the approximated KdV solution and the numerical solution.
基金The research of the first author is supported in part by NSFC grant 10572028Addi-tional support is provided by the National Basic Research Program of China under grant 2005CB321702+1 种基金by the Foundation of National Key Laboratory of Computational Physics under grant 9140C6902010603by the National Hi-Tech Inertial Confinement Fusion Committee of China.The research of the second author is supported in part by NSF grant DMS-0510345.
文摘Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics.The main purpose of this work is to demonstrate our claim in[3]that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges,which restricts the accuracy of the resulting scheme to at most second order.The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes.Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.
基金Item Sponsored by National Natural Science Foundation of China[No.1112521250936006]ITER Project[No.2009GB 104001]
文摘Based on an adaptive mesh refinement(AMR)technique,a multi-fluid MHD(magnetohydrodynamics)flow is directly simulated by using volume of fluid(VOF)method to capture the interface and charge conservation(consistent and conservative)scheme to calculate the Lorentz force.Effects of magnetic field on bubble rising in a liquid are conducted,in which the magnetic strength and magnetic directions decide the trajectory and the velocity of rising bubble. The history of kinetic distribution and trajectory is illustrated to describe the effect of the magnetic field on bubble driven flows.
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 National Natural Science Foundation of China(91130030).The authors are grateful to Dr.Jie Wu for his help on the manuscript.
文摘In this paper,a conservative fifth-order upwind compact scheme using centered stencil is introduced.This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric.Theoretical analysis shows that the proposed scheme is low-dissipative and has a relatively large stability range.To maintain the convergence rate of the whole spatial discretization,a proper non-periodic boundary scheme is also proposed.A detailed analysis shows that the spatial discretization implemented with the boundary scheme proposed by Pirozzoli[J.Comput.Phys.,178(2001),pp.81–117]is approximately fourth-order.Furthermore,a hybridmethodology,coupling the compact scheme with WENO scheme,is adopted for problems with discontinuities.Numerical results demonstrate the effectiveness of the proposed scheme.
基金This work was supported by the National Numerical Windtunnel Project NNW2019ZT4-B08Science Challenge Project TZZT2019-A2.3the National Natural Science Foundation of China Grant no.11871449.
文摘In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.
基金Sponsored partly by Priority-Scientific-Projects for China's 7th and 8th Five-Year Plana Priority Project of the Director's Foundation of the Institute of Atmospheric PhysicsChinese Academy of Sciences.
文摘In accordance with a new compensation principle of discrete computations,the traditional meteo- rological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy conservative fidelity schemes,thus resolving the problems of both nonlinear computa- tional instability and incomplete energy conservation,and raising the computational efficiency of the traditional schemes. As the numerical tests of the new schemes demonstrate,in solving the problem of energy conser- vation in operational computations,the new schemes can eliminate the (nonlinear) computational in- stability and,to some extent even the (nonlinear) computational diverging as found in the traditional schemes,Further contrasts between new and traditional schemes also indicate that,in discrete opera- tional computations,the new scheme in the case of nondivergence is capable of prolonging the valid in- tegral time of the corresponding traditional scheme,and eliminating certain kind of systematical com- putational“climate drift”,meanwhile increasing its computational accuracy and reducing its amount of computation.The working principle of this paper is also applicable to the problem concerning baroclin- ic primitive equations.
文摘In this paper, a numerical method is given to solve relativistic hydrodynamic equations with source terms by conservative finite difference scheme. In calculation, QGP (quark gluon plasma) phase transition is also considered. The numerical experiments have verified the effectiveness of the numerical method and some computational results are illustrated.
文摘A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.
