A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and t...A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).展开更多
In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-spar...In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high.We can obtain sparse matrices by applying compact schemes.In this article,we compare compact and exponential finite difference schemes of fourth order.The numerical solutions are calculated in quadruple precision(Real*16 or extended precision)in FORTRAN language,and iteratively obtained until reaching the round-off error magnitude around 1.0E−32.This procedure is performed to ensure that there is no iteration error.The Repeated Richardson Extrapolation(RRE)method combines numerical solutions in different grids,determining higher orders of accuracy.The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth,eighth,and tenth order of precision.The multigrid Full Approximation Scheme(FAS)is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.展开更多
A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can e...A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.展开更多
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform...In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.展开更多
In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the...In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).展开更多
In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gam...In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).展开更多
For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy...For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy in space, has long been desired. In the present work, a new kind of general parallel difference schemes for the nonlinear parabolic system is proposed. The general parallel difference schemes include, among others, two new parallel schemes. In one of them, to obtain the interface values on the interface of sub-domains an explicit scheme of Jacobian type is employed, and then the fully implicit scheme is used in the sub-domains. Here, in the explicit scheme of Jacobian type, the values at the points being adjacent to the interface points are taken as the linear combination of values of previous two time layers at the adjoining points of the inner interface. For the construction of another new parallel difference scheme, the main procedure is as follows. Firstly the linear combination of values of previous two time layers at the interface points among the sub-domains is used as the （Dirichlet） boundary condition for solving the sub-domain problems. Then the values in the sub-domains are calculated by the fully implicit scheme. Finally the interface values are computed by the fully implicit scheme, and in fact these calculations of the last step are explicit since the values adjacent to the interface points have been obtained in the previous step. The existence, uniqueness, unconditional stability and the second order accuracy of the discrete vector solutions for the parallel difference schemes are proved. Numerical results are presented to examine the stability, accuracy and parallelism of the parallel schemes.展开更多
Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and ma...Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.展开更多
In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both spac...In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.展开更多
文摘A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).
文摘In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high.We can obtain sparse matrices by applying compact schemes.In this article,we compare compact and exponential finite difference schemes of fourth order.The numerical solutions are calculated in quadruple precision(Real*16 or extended precision)in FORTRAN language,and iteratively obtained until reaching the round-off error magnitude around 1.0E−32.This procedure is performed to ensure that there is no iteration error.The Repeated Richardson Extrapolation(RRE)method combines numerical solutions in different grids,determining higher orders of accuracy.The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth,eighth,and tenth order of precision.The multigrid Full Approximation Scheme(FAS)is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.
基金Project supported by the National Natural Science Foundation of China (Nos. 10172015 and 90205010)
文摘A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.
文摘In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.
文摘In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).
文摘In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).
基金The project is supported by the Special Funds for Major State Basic Research Projects 2005CB321703, the National Nature Science Foundation of China （No. 10476002, 60533020）.
文摘For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy in space, has long been desired. In the present work, a new kind of general parallel difference schemes for the nonlinear parabolic system is proposed. The general parallel difference schemes include, among others, two new parallel schemes. In one of them, to obtain the interface values on the interface of sub-domains an explicit scheme of Jacobian type is employed, and then the fully implicit scheme is used in the sub-domains. Here, in the explicit scheme of Jacobian type, the values at the points being adjacent to the interface points are taken as the linear combination of values of previous two time layers at the adjoining points of the inner interface. For the construction of another new parallel difference scheme, the main procedure is as follows. Firstly the linear combination of values of previous two time layers at the interface points among the sub-domains is used as the （Dirichlet） boundary condition for solving the sub-domain problems. Then the values in the sub-domains are calculated by the fully implicit scheme. Finally the interface values are computed by the fully implicit scheme, and in fact these calculations of the last step are explicit since the values adjacent to the interface points have been obtained in the previous step. The existence, uniqueness, unconditional stability and the second order accuracy of the discrete vector solutions for the parallel difference schemes are proved. Numerical results are presented to examine the stability, accuracy and parallelism of the parallel schemes.
文摘Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.
基金Supported by the National Natural Science Foundation of China(No.10671060,No.10871061)the Youth Foundation of Hunan Education Bureau(No.06B037)the Construct Program of the Key Discipline in Hunan Province
文摘In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.
