The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded...The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.展开更多
A method for simulation of free surface problems is presented. Based on the viscous incompressible Navier-Stokes equations, space discretization of the flow is obtained by the least square finite element method. The t...A method for simulation of free surface problems is presented. Based on the viscous incompressible Navier-Stokes equations, space discretization of the flow is obtained by the least square finite element method. The time evolution is obtained by the finite difference method. Lagrangian description is used to track the free surface. The results are compared with the experimental dam break results, including water collapse in a 2D rectangular section and in a 3D cylinder section. A good agreement is achieved for the distance of surge front as well as the height of the residual column.展开更多
In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split i...In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.展开更多
A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element (CNLSMFE) formulation and proper orthogonal decomposition (POD) technique for two-dimensional (2D) Sobolev equat...A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element (CNLSMFE) formulation and proper orthogonal decomposition (POD) technique for two-dimensional (2D) Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.展开更多
We propose a new least squares finite element method to solve the Stokes problem with two sequential steps.The approximation spaces are constructed by the patch reconstruction with one unknown per element.For the firs...We propose a new least squares finite element method to solve the Stokes problem with two sequential steps.The approximation spaces are constructed by the patch reconstruction with one unknown per element.For the first step,we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace.By this space,we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure.In the second step,we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.We derive error estimates for all unknowns under both L 2 norms and energy norms.Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.展开更多
文摘The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.
基金Project supported by the National Natural Science Foundation of China (Nos.10302013,10572022)
文摘A method for simulation of free surface problems is presented. Based on the viscous incompressible Navier-Stokes equations, space discretization of the flow is obtained by the least square finite element method. The time evolution is obtained by the finite difference method. Lagrangian description is used to track the free surface. The results are compared with the experimental dam break results, including water collapse in a 2D rectangular section and in a 3D cylinder section. A good agreement is achieved for the distance of surge front as well as the height of the residual column.
基金financially supported by the National Natural Science Foundation of China(Grant No.51349011)the Foundation of Si’chuan Educational Committee(Grant No.17ZB0452)+1 种基金the Innovation Team Project of Si’chuan Educational Committee(Grant No.18TD0019)the Longshan Academic Talent Research Support Program of the Southwest of Science and Technology(Grant Nos.18LZX715 and 18LZX410)
文摘In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.
基金Supported by the National Natural Science Foundation of China(11271127)Science Research Projectof Guizhou Province Education Department(QJHKYZ[2013]207)
文摘A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element (CNLSMFE) formulation and proper orthogonal decomposition (POD) technique for two-dimensional (2D) Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.
基金supported by the Science Challenge Project(No.TZ2016002)the National Natural Science Foundation in China(No.11971041 and 11421101).
文摘We propose a new least squares finite element method to solve the Stokes problem with two sequential steps.The approximation spaces are constructed by the patch reconstruction with one unknown per element.For the first step,we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace.By this space,we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure.In the second step,we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.We derive error estimates for all unknowns under both L 2 norms and energy norms.Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.