A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation ...A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.展开更多
A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and d...A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and demonstrate the e?ciency of this approach.展开更多
The initial-boundary value problem of Burgers equation is considered. A prediction-correction Legendre spectral scheme is proposed. It possesses the accuracy of second order in time and higher order in space. The nume...The initial-boundary value problem of Burgers equation is considered. A prediction-correction Legendre spectral scheme is proposed. It possesses the accuracy of second order in time and higher order in space. The numerical experiments show the high accuracy of this approach.展开更多
In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were disc...In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.展开更多
In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis sh...In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.展开更多
This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artifi...This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.展开更多
In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caput...In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11771299,11371123)Natural Science Foundation of Henan Province(Grant No.202300410156).
文摘A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.
基金This work is supported in part by NSF of China, N.10471095, SF of Shanghai N.04JC14062, The Fund of ChineseEducation Ministry N.20040270002, The Shanghai Leading Academic Discipline Project N. T0401, The Funds forE-institutes of Universities N.E03004 and The special Funds for Major Specialities and N.04DB15 of ShanghaiEducation Commission.
文摘A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and demonstrate the e?ciency of this approach.
基金This work was supported in part by the Natural Science Foundation of China
文摘The initial-boundary value problem of Burgers equation is considered. A prediction-correction Legendre spectral scheme is proposed. It possesses the accuracy of second order in time and higher order in space. The numerical experiments show the high accuracy of this approach.
文摘In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.
基金supported partially by the innovation fund of Shanghai Normal Universitysupported partially by NSERC of Canada under Grant OGP0046726.
文摘In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.
文摘This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.
文摘In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.