In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average techniq...In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference scheme.The main work lies in the pointwise error estimate in H^(2)-norm.The optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded inequality.Moreover,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio restriction.Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.展开更多
The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, wher...The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.展开更多
Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations. For time discretization, a three-stage explicit Runge-Kutta method with TVD property is used at pr...Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations. For time discretization, a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage, a cubic spline function is adopted at correcting stage, which made the time discretization accuracy up to fourth order; For spatial discretization, a three-point explicit compact difference scheme with arbitrary order accuracy is employed. The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme. The numerical results agree well with the experimental data. At the same time, the comparisons of the two numerical results between the present scheme and low accuracy difference method are made, which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations. As a valid sample, the wave propagation on the rectangular step is formulated by the present scheme, the modelled results are in better agreement with the experimental data than those of Kittitanasuan.展开更多
A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite differen...A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.展开更多
A second-order mixing difference scheme with a limiting factor is deduced with the reconstruction gradient method and applied to discretizing the Navier-Stokes equation in an unstructured grid.The transform of nonorth...A second-order mixing difference scheme with a limiting factor is deduced with the reconstruction gradient method and applied to discretizing the Navier-Stokes equation in an unstructured grid.The transform of nonorthogonal diffusion items generated by the scheme in discrete equations is provided.The Delaunay triangulation method is improved to generate the unstructured grid.The computing program based on the SIMPLE algorithm in an unstructured grid is compiled and used to solve the discrete equations of two types of incompressible viscous flow.The numerical simulation results of the laminar flow driven by lid in cavity and flow behind a cylinder are compared with the theoretical solution and experimental data respectively.In the former case,a good agreement is achieved in the main velocity and drag coefficient curve.In the latter case,the numerical structure and development of vortex under several Reynolds numbers match well with that of the experiment.It is indicated that the factor difference scheme is of higher accuracy,and feasible to be applied to Navier-Stokes equation.展开更多
The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain th...The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.展开更多
Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been...Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.展开更多
By comparing the energy spectrum and total kinetic energy, the effects of numerical errors （which arise from aliasing and discretization errors）, subgrid-scale （SGS） models, and their interactions on direct numeri...By comparing the energy spectrum and total kinetic energy, the effects of numerical errors （which arise from aliasing and discretization errors）, subgrid-scale （SGS） models, and their interactions on direct numerical simulation （DNS） and large eddy simulation （LES） are investigated, The decaying isotropic turbulence is chosen as the test case. To simulate complex geometries, both the spectral method and Pade compact difference schemes are studied. The truncated Navier-Stokes （TNS） equation model with Pade discrete filter is adopted as the SGS model. It is found that the discretization error plays a key role in DNS. Low order difference schemes may be unsuitable. However, for LES, it is found that the SGS model can represent the effect of small scales to large scales and dump the numerical errors. Therefore, reasonable results can also be obtained with a low order discretization scheme.展开更多
A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and...A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.展开更多
Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlin...Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).展开更多
The authors' proposed averaging finite difference(AFD) theme bad been tested through the numerical example of a 2-D driven cavity flow, and its accuracy and feasibility have been indicated [1]. In this paper, the ...The authors' proposed averaging finite difference(AFD) theme bad been tested through the numerical example of a 2-D driven cavity flow, and its accuracy and feasibility have been indicated [1]. In this paper, the scheme was adopted to simulate the high Re flow with irregular boundaries so as to prove that it is also successful for this scheme to be used in the flow with irregular boundaries. A 2-D flow around a semicircular obstruction was taken as the calculation example and the first order wall vorticity formula for irregular mesh derived by the authors in Ref. [2] was Chosen to calculate the wall vorticity in the considered example. The salutations were carried out under the conditions: Re = 43, 100, 300, 500, 1000 and 10000.The reasonable flow results and the curve of separated angle or reattached length VS Re were given. The results have shown that the AFD scheme can be extended to the simulations of separated flow with arbitrary boundaries and at high Reynolds number really.展开更多
Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlin...Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).展开更多
In this paper a simple and efficient implicit finite-difference scheme is used for depth-averaged two-dimensional storm surge model. This finite-difference scheme is simpler and more efficient than the widely-used ADI...In this paper a simple and efficient implicit finite-difference scheme is used for depth-averaged two-dimensional storm surge model. This finite-difference scheme is simpler and more efficient than the widely-used ADI scheme. Accuracy analysis and stability analysis indicate that the scheme has two-order accuracy and is unconditionally stable when the grid size is constant. The present analysis results show that the scheme is of higher numerical accuracy than that introduced by Maa (1990). After tested by ideal models, a calculation example of a real typhoon surge is carried out, the results of the numerical simulation coincide well with the observed data and the accuracy is sufficient for engineering applications.展开更多
地震勘探中的波动方程正演模拟受计算和存储能力的限制,只能在有限空间进行,需要对其设置边界条件。常用的完全匹配层(Perfect Match Layer,PML)是一种应用广泛的边界条件,需要对边界条件设定一定的层数,层数太大时会降低正演速度和增...地震勘探中的波动方程正演模拟受计算和存储能力的限制,只能在有限空间进行,需要对其设置边界条件。常用的完全匹配层(Perfect Match Layer,PML)是一种应用广泛的边界条件,需要对边界条件设定一定的层数,层数太大时会降低正演速度和增大内存存储。因此,文中通过将PML边界条件和改进后的梯度黏弹边界(Improved Gradient Viscoelastic Layer,IGVL)结合提出一种PML-IGVL边界条件,同时,为了减少截断误差和数值频散,将PML-IGVL边界条件应用于紧致交错网格的黏滞声波方程中进行数值模拟。均匀介质和Marmousi模型中的波场数值模拟结果表明,相对于PML边界条件,PML-IGVL边界条件使用的边界厚度更小,在相同较低边界层数下,能更好地吸收反射波,同时能节省更多的存储空间,并提高正演效率,证明了PML-IGVL边界条件的有效性和优越性,是正演模拟中一种有效的边界吸收方法。展开更多
基金supported in part by Natural Sciences Foundation of Zhejiang Province(No.LZ23A010007)in part by the National Natural Science Foundation of China(No.12271518)+1 种基金Natural Science Foundation of Jiangsu Province(No.BK20201149)the Fundamental Research Funds of Xuzhou(No.KC21019)
文摘In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference scheme.The main work lies in the pointwise error estimate in H^(2)-norm.The optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded inequality.Moreover,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio restriction.Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471262)Henan University of Technology High-level Talents Fund,China(Grant No.2018BS039)
文摘The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.
基金The project was financially supported by the National Natural Science Foundation of China (Grant No50479053)
文摘Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations. For time discretization, a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage, a cubic spline function is adopted at correcting stage, which made the time discretization accuracy up to fourth order; For spatial discretization, a three-point explicit compact difference scheme with arbitrary order accuracy is employed. The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme. The numerical results agree well with the experimental data. At the same time, the comparisons of the two numerical results between the present scheme and low accuracy difference method are made, which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations. As a valid sample, the wave propagation on the rectangular step is formulated by the present scheme, the modelled results are in better agreement with the experimental data than those of Kittitanasuan.
基金the National Natural Science Foundation of China
文摘A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.
基金Supported by National Natural Science Foundation of China (No. 10632050)
文摘A second-order mixing difference scheme with a limiting factor is deduced with the reconstruction gradient method and applied to discretizing the Navier-Stokes equation in an unstructured grid.The transform of nonorthogonal diffusion items generated by the scheme in discrete equations is provided.The Delaunay triangulation method is improved to generate the unstructured grid.The computing program based on the SIMPLE algorithm in an unstructured grid is compiled and used to solve the discrete equations of two types of incompressible viscous flow.The numerical simulation results of the laminar flow driven by lid in cavity and flow behind a cylinder are compared with the theoretical solution and experimental data respectively.In the former case,a good agreement is achieved in the main velocity and drag coefficient curve.In the latter case,the numerical structure and development of vortex under several Reynolds numbers match well with that of the experiment.It is indicated that the factor difference scheme is of higher accuracy,and feasible to be applied to Navier-Stokes equation.
文摘The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.
文摘Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.
基金Project supported by the National Natural Science Foundation of China (No.10502029)the Scientific Research Foundation for Returned Overseas Chinese Scholars of Ministry of Education of China
文摘By comparing the energy spectrum and total kinetic energy, the effects of numerical errors （which arise from aliasing and discretization errors）, subgrid-scale （SGS） models, and their interactions on direct numerical simulation （DNS） and large eddy simulation （LES） are investigated, The decaying isotropic turbulence is chosen as the test case. To simulate complex geometries, both the spectral method and Pade compact difference schemes are studied. The truncated Navier-Stokes （TNS） equation model with Pade discrete filter is adopted as the SGS model. It is found that the discretization error plays a key role in DNS. Low order difference schemes may be unsuitable. However, for LES, it is found that the SGS model can represent the effect of small scales to large scales and dump the numerical errors. Therefore, reasonable results can also be obtained with a low order discretization scheme.
文摘A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.
基金Project of Knowledge Innovation Program from Chinese Academy of Sciences (KZCX2-109).
文摘Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).
文摘The authors' proposed averaging finite difference(AFD) theme bad been tested through the numerical example of a 2-D driven cavity flow, and its accuracy and feasibility have been indicated [1]. In this paper, the scheme was adopted to simulate the high Re flow with irregular boundaries so as to prove that it is also successful for this scheme to be used in the flow with irregular boundaries. A 2-D flow around a semicircular obstruction was taken as the calculation example and the first order wall vorticity formula for irregular mesh derived by the authors in Ref. [2] was Chosen to calculate the wall vorticity in the considered example. The salutations were carried out under the conditions: Re = 43, 100, 300, 500, 1000 and 10000.The reasonable flow results and the curve of separated angle or reattached length VS Re were given. The results have shown that the AFD scheme can be extended to the simulations of separated flow with arbitrary boundaries and at high Reynolds number really.
文摘Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).
文摘In this paper a simple and efficient implicit finite-difference scheme is used for depth-averaged two-dimensional storm surge model. This finite-difference scheme is simpler and more efficient than the widely-used ADI scheme. Accuracy analysis and stability analysis indicate that the scheme has two-order accuracy and is unconditionally stable when the grid size is constant. The present analysis results show that the scheme is of higher numerical accuracy than that introduced by Maa (1990). After tested by ideal models, a calculation example of a real typhoon surge is carried out, the results of the numerical simulation coincide well with the observed data and the accuracy is sufficient for engineering applications.
