In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square...In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.展开更多
We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. ...We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.展开更多
In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularit...In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.展开更多
We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artific...We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.展开更多
基金Supported by the NSF of the Higher Education Institutions of Jiangsu Province(10KJD110006)Supported by the grant of Jiangsu Institute of Education(Jsjy2009zd03)Supported by the Qing Lan Project of Jiangsu Province(2010)
文摘In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.
文摘We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.
基金supported by the National Natural Science Foundation of China(Grant Nos.11871467,11471329).
文摘In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.
基金Singapore A*STAR SERC PSF-Grant No.1321202067National Natural Science Foundation of China Grant NSFC41390452the Doctoral Programme Foundation of Institution of Higher Education of China as well as by the Austrian Science Foundation(FWF)under grant No.F41(project VICOM)and grant No.I830(project LODIQUAS)and grant No.W1245 and the Austrian Ministry of Science and Research via its grant for the WPI.
文摘We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.
