双币种期权定价模型的一个新ADI并行差分方法

双币种期权是一种重要的金融衍生产品,其定价模型是一个含有混合导数项的二维Black-Scholes方程,研究它的数值解法有着非常重要的理论意义和实际价值.本文给出求解双币种期权定价模型的基于Craig-Sneyd分裂法的一个新ADI差分方法(C-S ADI),该方法首先将二维B1ack-Scholes方程分裂为两个一维方程和一个含有混合导数的二维方程,然后分别对一维方程构造半隐式格式,对含混合导数的二维方程构造显式格式进行计算.C-S ADI差分方法具有以下优点:并行性,无条件稳定性,收敛性及空间二阶、时间一阶的计算精度.理论分析与数值试验表明,相比于经典的Crank-Nicolson差分格式和已有的基于Douglas Rachford分裂法的ADI差分格式(D-R ADI),本文格式计算精度更高,并且由于其具有天然的并行特性,本文格式比串行的Crank-Nicolson格式节省了近1/5的计算时间,证实了该方法对求解双币种期权定价模型是有效的.     

一类二维抛物方程反问题的CCD-ADI方法

许多工程问题可通过带有未知参数的抛物方程求解.因此,发展高精度数值方法求解这类反问题非常重要.本文提出一种交替方向隐格式(ADI)的三层线性化组合紧致差分(CCD)格式求解带控制参数的二维非定常反应扩散方程.该方法在时间上达到二阶精度,空间上达到六阶精度.在每个ADI迭代步,只需求解一个块三对角系统,可通过块Thomas算法快速求解.此外,我们严格证明在周期性边界条件下,CCD-ADI方法解的存在性和唯一性.最后,通过与已有空间四阶方法对比,用数值算例验证新方法的无条件稳定性、精度与效率.     

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.     

Bates模型下一种美式期权高阶紧致有限差分定价方法

基于Jain提出的高阶紧致有限差分格式(high order compact of Jain,HOCJ),结合卷积积分(convolution integral)与快速傅里叶变换(FFT),构建了一种新颖的数值方法,简称HOCJ-CF,并用于Bates模型下美式看跌期权定价.针对期权定价偏积分微分方程(PIDE)的微分项,首先将其拆分成三个子偏微分方程(sub-PDE),然后分别应用Numerov离散方法,衍生出具有空间四阶精度和时间二阶精度的HOCJ格式;积分项则将其转化成卷积积分,并运用FFT.在相同模型参数设置下,数值结果验证了新方法在精度、收敛率及效率相比IMEX格式的优越性.     

三维对流扩散方程的三种高精度分裂格式

在算子分裂法思想的基础上,将两种高精度的离散格式推广应用于三维对流扩散方程,同时对经典ADI格式的对流项做了改进,改进后的格式的对流项对空间具有4阶精度,而经典ADI格式对空间只有2阶精度,由此可见,提高了该格式的实用性．最后对两种典型的浓度场进行了数值模拟,将3种格式的计算结果与解析解以及其它传统差分格式的计算结果进行了对比,得出当Peclet数不大于5时,3种格式均获得了令人满意的数值结果,说明推广的这三种方法具有很高的准确性和可靠性．     

求解时间分布阶扩散方程的两个高阶有限差分格式

基于复化Simpson公式和复化两点Gauss-Legendre公式，构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式，这两种格式在时间方向都具有三阶精度，而在分布阶和空间方向可达到四阶精度.数值结果表明，两种算法都是稳定且收敛的，从而是有效的.两种格式的收敛速率也通过数值实验进行了验证，并且通过和文献中的算法对比可以得出其更为高效,     

Lagrange非结构网格高阶交错型守恒气体动力学格式

提出Lagrange(拉格朗日)非结构网格高阶交错型守恒气体动力学格式.用产生于当前时刻子网格密度和网格声速的子网格压力和MUSCL方法构造了高阶子网格力,利用高阶子网格力构造了高阶空间通量,借助时间中点通量的Taylor(泰勒)展开完成了高阶时间通量离散.研制了Lagrange非结构网格高阶交错型守恒气体动力学格式.对Saltzman活塞问题等进行了数值模拟,数值结果显示了Lagrange非结构网格高阶交错型守恒气体动力学格式的有效性和精确性.     

求解对流扩散方程的低耗散中心迎风格式

以四阶CWENO重构为基础,通过将对流项采用低耗散中心迎风格式离散,扩散项采用四阶中心差分格式离散,对得到的半离散格式采用四阶龙格库塔方法在时间方向上推进,得到一种求解对流扩散方程的高阶有限差分格式.数值结果验证了该格式的四阶精度和基本无振荡特性.     

高阶特征线性及其与semi-Lagrangian方法的比较

特征线性与semi-Lagrangian方法都是处理流体方程时间离散的两种有效的方法．它们比经典的半隐格式，如Backward-Euler／Adams-Bashforth方法有更好的稳定性．本提出一种基于高阶空间离散的特征线法，通过稳定性，精度和计算复杂性与semi-Lagrangian方法进行比较，分析了高阶特征线法的有效性和适用性，并从数值试验上对分析结果进行验证．     

无界区域上波导问题的有效谱迦略金方法(英文)

旨在求解二维无界区域上的波导问题.通过在人工边界施加精确非反射边界条件(nonreflecting boundary condition,NRBC),可以将无界问题截断为有界问题,然后,给出了在空间上用傅里叶谱迦略金方法、时间上用Newmark格式和Richardson外插技巧的数值格式来求解该有界问题.介绍了一个快速的时间行进格式来处理包含在NRBC中的时间方向的卷积.提出的算法在空间和时间上都可以达到高阶精度.另外,也证明了该截断问题的适定性.     

An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(τ² + h⁶) accurate, where τ, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.     

The new alternating direction implicit difference methods for the wave equations

A new second-order alternating direction implicit (ADI) scheme, based on the idea of the operator splitting, is presented for solving two-dimensional wave equations. The scheme is also extended to a high-order compact difference scheme. Both of them have the advantages of unconditional stability, less impact of the perturbing terms on the accuracy, and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and costs less in computational time. Numerical examples are presented and the results are very satisfactory.     

The new alternating direction implicit difference methods for solving three-dimensional parabolic equations

A new alternating direction implicit (ADI) scheme for solving three-dimensional parabolic equations with nonhomogeneous boundary conditions is presented. The scheme is also extended to high-order compact difference scheme. Both of them have the advantages of unconditional stability and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and uses less computational time. Numerical examples are presented and the results are very satisfactory.     

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

In this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems

In this article, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank–Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection–diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection–diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods. The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.     

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations北大核心CSCD

针对三维非稳态对流扩散反应方程,构造了一种高精度紧致有限差分格式,对空间的离散采用四阶紧致差分方法,对时间的离散采用Taylor级数展开和余项修正技术,所提格式在时间上的精度为二阶、在空间上的精度为四阶。利用Fourier稳定性分析法证明了该格式是无条件稳定的。最后给出数值算例验证了理论结果。     

Optimization via Chebyshev polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices. In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.