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
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
The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtai...The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtain a weak formulation of heterogenous viscoacoustic wave propagation in an infinite domain,the viscoacoustic medium is first characterized by its frequency-dependent complex bulk compliance instead of the classically used complex bulk modulus.Then,a mechanical model using serially connected standard linear solids(SSLS)is built to obtain the rational approximation of the complex bulk compliance whose parameters are calculated via an adapted nonlinear optimization method.Utilizing the obtained bulk compliance-based constitutive relation,a novel second-order viscoacoustic wave equation in the frequency domain is derived,of which the weak formulation can be physically explained as the virtual work equation and can thus be discretized using a continuous spectral element method in space.Additionally,a new method is introduced to address the convolution terms involved in the inverse Fourier transform,whose accurate time integration can then be achieved using an explicit time scheme,which avoids the transient growth that exists in the classical method.The resulting full time-space decoupling scheme can handle wave propagation in arbitrary heterogeneous media.Moreover,to treat the wave propagation in an infinite domain,a perfectly matched layer in weak formulation is derived for the truncation of the infinite domain via complex coordinate stretching of the virtual work equation.With only minor modification,the resulting perfectly matched layer can be implemented using the same time scheme as for the wave equation inside the truncated domain.The accuracy,numerical stability,and versatility of the new proposed scheme are demonstrated with numerical examples.展开更多
Nonlinear problems widely exist in many aspects of the natural field.The nonlinear situation makes it difficult for most existing solvers to deal with.Therefore,constructing an efficient and accurate solver is a chall...Nonlinear problems widely exist in many aspects of the natural field.The nonlinear situation makes it difficult for most existing solvers to deal with.Therefore,constructing an efficient and accurate solver is a challenge.In this paper,a Legendre spectral method is developed for the nonlinear Volterra integrodifferential equation.The error analysis is also provided to justify the spectral rate of convergence for the er-rors of approximate solution and approximate derivative decay exponentially in both the L2 norm and the infinity norm.In the end,numerical results are displayed to con-firm the effectiveness of the Legendre spectral analysis.展开更多
In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem...In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.展开更多
In this work,we concern with the numerical approach for delay differential equations with random coefficients.We first show that the exact solution of the problem considered admits good regularity in the random space,...In this work,we concern with the numerical approach for delay differential equations with random coefficients.We first show that the exact solution of the problem considered admits good regularity in the random space,provided that the given data satisfy some reasonable assumptions.A stochastic collocation method is proposed to approximate the solution in the random space,and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations.Convergence property of the proposed method is analyzed.It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space.Numerical examples are given to illustrate the theoretical results.展开更多
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.
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National Natural Science Foundation of China under Grant No.U2039209the National Key R&D Program of China under Grant No.2022YFC3004303+1 种基金the Heilongjiang Natural Science Foundation for Distinguished Young Scholars under Grant No.JQ2022E006Heilongjiang Natural Science Foundation Joint Guidance Project under Grant No.LH2021E122。
文摘The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtain a weak formulation of heterogenous viscoacoustic wave propagation in an infinite domain,the viscoacoustic medium is first characterized by its frequency-dependent complex bulk compliance instead of the classically used complex bulk modulus.Then,a mechanical model using serially connected standard linear solids(SSLS)is built to obtain the rational approximation of the complex bulk compliance whose parameters are calculated via an adapted nonlinear optimization method.Utilizing the obtained bulk compliance-based constitutive relation,a novel second-order viscoacoustic wave equation in the frequency domain is derived,of which the weak formulation can be physically explained as the virtual work equation and can thus be discretized using a continuous spectral element method in space.Additionally,a new method is introduced to address the convolution terms involved in the inverse Fourier transform,whose accurate time integration can then be achieved using an explicit time scheme,which avoids the transient growth that exists in the classical method.The resulting full time-space decoupling scheme can handle wave propagation in arbitrary heterogeneous media.Moreover,to treat the wave propagation in an infinite domain,a perfectly matched layer in weak formulation is derived for the truncation of the infinite domain via complex coordinate stretching of the virtual work equation.With only minor modification,the resulting perfectly matched layer can be implemented using the same time scheme as for the wave equation inside the truncated domain.The accuracy,numerical stability,and versatility of the new proposed scheme are demonstrated with numerical examples.
基金supported by the Visiting scholar program of National Natural Science Foundation of China(Nos.12426616 and 12271223)Natural Science Research Startup Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(No.NY223127)+1 种基金the Research Projects of Guangdong Provincial Education Department(No.2021KTSCX071,2022KTSCX077,and HSGDJG21356-372)Hanshan Normal University project(No.QD202212).
文摘Nonlinear problems widely exist in many aspects of the natural field.The nonlinear situation makes it difficult for most existing solvers to deal with.Therefore,constructing an efficient and accurate solver is a challenge.In this paper,a Legendre spectral method is developed for the nonlinear Volterra integrodifferential equation.The error analysis is also provided to justify the spectral rate of convergence for the er-rors of approximate solution and approximate derivative decay exponentially in both the L2 norm and the infinity norm.In the end,numerical results are displayed to con-firm the effectiveness of the Legendre spectral analysis.
基金the National Basic Research Programthe National Natural Science Foundation of China(Grant No.2005CB321703)+2 种基金Scientific Research Fund of Hunan Provincial Education Departmentthe Outstanding Youth Scientist of the National Natural Science Foundation of China(Grant No.10625106)the National Basic Research Program of China(Grant No.2005CB321701)
文摘In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.
基金the National Natural Science Foundation of China(No.91130003 and No.11201461).
文摘In this work,we concern with the numerical approach for delay differential equations with random coefficients.We first show that the exact solution of the problem considered admits good regularity in the random space,provided that the given data satisfy some reasonable assumptions.A stochastic collocation method is proposed to approximate the solution in the random space,and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations.Convergence property of the proposed method is analyzed.It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space.Numerical examples are given to illustrate the theoretical results.
