摘要：
The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrodinger equation in the semiclassical regime, where the Planck constant epsilon is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrodinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O(max{Delta t(3), Delta t(5)/epsilon}), while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O(Delta t(2)/epsilon) and O(Delta t(3)/epsilon), respectively, where Delta t is the time step-size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step-size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrodinger equation with a quadratic potential. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
摘要：
In this paper, we consider the fast numerical valuation of European and American options under Merton’s jump-diffusion model, which is given by a partial integro-differential equations. Due to the singularities and discontinuities of the model, the time-space grids are nonuniform with refinement near the strike price and expiry. On such nonuniform grids, the spatial differential operators are discretized by finite difference methods, and time stepping is performed using the discontinuous Galerkin finite element method. Owing to the nonuniform grids, algebraic multigrid method is used for solving the dense algebraical system resulting from the discretization of the integral term associated with jumps in models, which is more challenging. Numerical comparison of algebraic multigrid, the generalized minimal residual method, and the incomplete LU preconditioner shows that algebraic multigrid method is superior to and more effective than the other two methods in solving such dense algebraical system.
摘要：
In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.
摘要：
In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre-Gauss-Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method.
摘要：
This paper is concerned with the stability of the one-leg methods for nonlinear Volterra functional differential equations (VFDEs). The contractivity and asymptotic stability properties are first analyzed for quasi-equivalent and A-stable one-leg methods by use of two lemmas proven in this paper. To extend the analysis to the case of strongly A-stable one-leg methods, Nevanlinna and Liniger's technique of introducing new norm is used to obtain the conditions for contractivity and asymptotic stability of these methods. As a consequence, it is shown that one-leg theta-methods (theta is an element of(1/2, 1]) with linear interpolation are unconditionally contractive and asymptotically stable, and 2-step Adams type method and 2-step BDF method are conditionally contractive and asymptotically stable. The bounded stability of the midpoint rule is also proved with, the help of the concept of semi-equivalent one-leg methods. (C) 2017 Elsevier Inc. All rights reserved.
摘要：
Motivated by recent stability results on one-step methods, especially Runge-Kutta methods, for the generalized pantograph equation (GPE), in this paper we study the stability of one-leg multistep methods for these equations since the one-leg methods have less computational cost than Runge-Kutta methods. To do this, a new stability concept, Gq((q) over bar)-stability defined for variable stepsizes one-leg methods with the stepsize ratio q which is an extension of G-stability defined for constant stepsizes one-leg methods, is introduced. The Lyapunov functional of linear system is obtained and numerically approximated. It is proved that a Gg((q) over bar)-stable fully-geometric mesh one-leg method can preserve the decay property of the Lyapunov functional for any q epsilon [1,(q) over bar]. The asymptotic contractivity, a new stability concept at vanishing initial interval, is introduced for investigating the effect of the initial interval approximation on the stability of numerical solutions. This property and the bounded stability of Gq((q) over bar)-stable one-leg methods for linear and nonlinear problems are analyzed. A numerical example which further illustrates our theoretical results is provided. (C) 2017 IMACS. Published by Elsevier B.V. All rights reserved.
