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
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
In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed....In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.展开更多
Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refrac...Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations. Many methods used to solve coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.展开更多
The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potent...We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.展开更多
In this paper, a compact finite difference method is presented for solving the initial boundary value problems for the improved Boussinesq equation with damping terms. The fourth-order equation can be transformed into...In this paper, a compact finite difference method is presented for solving the initial boundary value problems for the improved Boussinesq equation with damping terms. The fourth-order equation can be transformed into a first-order ordinary differential system, and then, the classical Pad4 approximation is used to discretize spatial derivative in the non- linear partial differential equations. The resulting coefficient matrix for the semi-discrete scheme is tri-diagonal and can be solved efficiently. In order to maintain the same order of convergence, the classical fourth-order Runge-Kutta method is the preferred method for explicit time integration. Soliton-type solutions are used to evaluate the accuracy of the method, and various numerical experiments are designed to test the different effects of the damping terms.展开更多
We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms.The meth...We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms.The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented,thus arriving at a compact formulation.In general,a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case.The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations.The convergence of the scheme has been established using the matrix analysis,and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix.The difference scheme has been applied to test convection diffusion equation,steady state Burger’s equation,ocean model and a semi-linear elliptic equation.The computational results confirm the theoretical order and accuracy of the method.展开更多
In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solutio...In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solution is considered,which often generates a singular source and increases the difficulty of numerically solving the equation.The Crank-Nicolson technique,combined with the midpoint formula and the second-order convolution quadrature formula,is used for the time discretization.To increase the spatial accuracy,a fourth-order compact difference approximation,which is constructed by two compact difference operators,is adopted for spatial discretization.Then,the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space.Finally,numerical experiments are given to support our theoretical results.展开更多
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.
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supported by the Project for the National Natural Science Foundation of China(No.12261103).
文摘In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.
文摘Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations. Many methods used to solve coupled nonlinear Schr<span style="white-space:nowrap;">ödinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.
文摘The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
基金supported by Ministry of Education of Singapore grant R-146-000-120-112the National Natural Science Foundation of China(Grant No.11131005)the Doctoral Programme Foundation of Institution of Higher Education of China(Grant No.20110002110064).
文摘We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.
文摘In this paper, a compact finite difference method is presented for solving the initial boundary value problems for the improved Boussinesq equation with damping terms. The fourth-order equation can be transformed into a first-order ordinary differential system, and then, the classical Pad4 approximation is used to discretize spatial derivative in the non- linear partial differential equations. The resulting coefficient matrix for the semi-discrete scheme is tri-diagonal and can be solved efficiently. In order to maintain the same order of convergence, the classical fourth-order Runge-Kutta method is the preferred method for explicit time integration. Soliton-type solutions are used to evaluate the accuracy of the method, and various numerical experiments are designed to test the different effects of the damping terms.
文摘We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms.The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented,thus arriving at a compact formulation.In general,a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case.The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations.The convergence of the scheme has been established using the matrix analysis,and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix.The difference scheme has been applied to test convection diffusion equation,steady state Burger’s equation,ocean model and a semi-linear elliptic equation.The computational results confirm the theoretical order and accuracy of the method.
基金supported by Natural Science Foundation of Jiangsu Province of China(Grant No.BK20201427)National Natural Science Foundation of China(Grant Nos.11701502 and 11871065)。
文摘In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solution is considered,which often generates a singular source and increases the difficulty of numerically solving the equation.The Crank-Nicolson technique,combined with the midpoint formula and the second-order convolution quadrature formula,is used for the time discretization.To increase the spatial accuracy,a fourth-order compact difference approximation,which is constructed by two compact difference operators,is adopted for spatial discretization.Then,the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space.Finally,numerical experiments are given to support our theoretical results.
