二维系数不连续Helmholtz方程的高阶紧致差分格式

针对二维系数不连续Helmholtz方程,提出和研究了高阶紧致差分格式,在波数跳跃位置引入局部网格加密技巧进行网格加密.数值实验验证,该高阶紧致差分格式用于求解二维系数不连续Helmholtz方程可以达到四阶精度,局部网格加密技巧能够有效地提高数值解的精度.     

带非局部边界条件的非线性抛物方程的隐式差分解法

在准静态弹性力学中常遇到求解带有非局部边界条件的抛物方程初边值问题.本文构造了一个数值求解带有非局部边界条件的非线性抛物方程的隐式差分格式,利用离散泛函分析的知识和不动点定理证明了差分解是存在的,且在离散最大模意义下关于时间步长一阶收敛,关于空间步长二阶收敛,并给出了数值算例.     

欧氏看涨期权定价问题的一种有效七点差分GMRES方法

用有限差分方法研究欧氏看涨期权定价问题.首先,将Black-Scholes方程通过等价代换化成一个标准的抛物型偏微分方程.其次,在求解区域构造时间精度为O（△τ^3）、空间精度为O（h^6）的差分格式,并通过Fourier分析方法证明该差分格式是无条件稳定的;边界区域选用精度较高、稳定性好的Crank-Nicolson格式,建立迭代方程.然后,用GMRES（generalized minimal residual）方法求解该方法.最后,给出一个欧氏看涨期权的数值算例,并与解析解进行比较,验证差分格式的有效性.     

两点混合边值问题的紧有限体积格式

本文针对常系数和变系数两点混合边值问题提出一种紧有限体积格式,该格式形成的线性代数方程组具有三对角性质,可以使用追赶法求解.证明格式按照H1半范数具有四阶收敛精度.利用节点计算值,给出单元中点值和一阶导数值的高精度后处理计算公式,这两个公式同样具有四阶精度.数值算例验证了理论分析的正确性,并说明了格式的有效性.     

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.     

非线性Black-Scholes模型下算术平均亚式期权定价问题

在非线性Black-Scholes模型下,研究了算术平均亚式期权定价问题.首先利用单参数摄动方法,将亚式期权适合的偏微分方程分解成一系列常系数抛物方程.其次通过计算这些常系数抛物型方程的解,给出了算术平均亚式期权的近似定价公式.最后分析了近似结论的误差估计,并通过数值算例验证了所得近似结论的合理性.     

带跳随机波动率模型的高阶ADI分裂格式

针对带跳随机波动率模型满足的偏积分微分方程,提出一种新的高阶交替方向隐式（ADI）有限差分格式,该模型是一个具有混合导数和非常数系数的对流扩散型初边值问题.我们将不同的高阶空间离散与时间步ADI分裂格式相结合,得到了一种空间四阶精度、时间二阶精度的有效方法,并采用Fourier方法分析了高阶ADI格式的稳定性.最后,通过对欧式看跌期权定价模型进行数值实验证实了数值方法的高阶收敛性.     

美式期权定价问题的变网格差分方法

本文提出一种求解美式期权定价自由边值问题的变网格差分方法.通过建立一个自由边界所满足的方程,利用变网格技术可同时求出期权的差分解和最佳执行边界.本文分别讨论了显式和隐式变网格差分格式,并给出了差分解的收敛性和稳定性分析.数值实验表明本文算法是一个非常有效的期权定价算法.     

二维抛物型方程的高精度多重网格解法

提出了数值求解二维抛物型方程的一种新的高精度加权平均紧隐格式，利用Fourier分析方法证明了该格式是无条件稳定的，为了克服传统迭代法在求解隐格式是收敛速度慢的缺陷，利用了多重网格加速技术，大大加快了迭代收敛速度，提高了求解效率，数值实验结果验证了方法的精确性和可靠性。     

非线性Black-Scholes模型下Bala期权定价

在非线性Black-Scholes模型下,研究了Bala期权定价问题.首先利用双参数摄动方法,将Bala期权适合的偏微分方程分解成一系列常系数抛物方程.其次通过计算这些常系数抛物型方程的解,给出了Bala期权的近似定价公式.最后利用Green函数分析了近似结论的误差估计.     

CONSERVATION OF THREE—POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

For nonlinear hyperbloic problems,Conservation of the numerical scheme is important for convergence to the correct weak solutions.In this paper the the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied,and a conservative interface treatment is derived for compact schemes on patched grids .For a pure initial value problem,the compact scheme is shown to be equivalent to a scheme in the usual conservative form .For the case of a mixed initial boundary value problem,the compact scheme is conservative only if the rounding errors are small enough.For a pactched grid interface,a conservative interface condition useful for mesh fefiement and for parallel computation is derived and its order of local accuracy is analyzed.     

Explicit local time-stepping methods for Maxwell’s equations

Explicit local time-stepping methods are derived for time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretisation in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fully explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leap-frog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes.     

Modified equation based mesh adaptation algorithm for evolutionary scalar partial differential equations

It is well known that on uniform mesh classical higher order schemes for evolutionary problems yield an oscillatory approximation of the solution containing discontinuity or boundary layers. In this article, an entirely new approach for constructing locally adaptive mesh is given to compute nonoscillatory solution by representative “second” order schemes. This is done using modified equation analysis and a notion of data dependent stability of schemes to identify the solution regions for local mesh adaptation. The proposed algorithm is applied on scalar problems to compute the solution with discontinuity or boundary layer. Presented numerical results show underlying second order schemes approximate discontinuities and boundary layers without spurious oscillations.     

Fourth‐order compact scheme with local mesh refinement for option pricing in jump‐diffusion model

The value of a contingent claim under a jump‐diffusion process satisfies a partial integro‐differential equation. A fourth‐order compact finite difference scheme is applied to discretize the spatial variable of this equation. It is discretized in time by an implicit‐explicit method. Meanwhile, a local mesh refinement strategy is used for handling the nonsmooth payoff condition. Moreover, the numerical quadrature method is exploited to evaluate the jump integral term. It guarantees a Toeplitz‐like structure of the integral operator such that a fast algorithm is feasible. Numerical results show that this approach gives fourth‐order accuracy in space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

An IMEX‐BDF2 compact scheme for pricing options under regime‐switching jump‐diffusion models

In this paper, an implicit‐explicit two‐step backward differentiation formula (IMEX‐BDF2) together with finite difference compact scheme is developed for the numerical pricing of European and American options whose asset price dynamics follow the regime‐switching jump‐diffusion process. It is shown that IMEX‐BDF2 method for solving this system of coupled partial integro‐differential equations is stable with the second‐order accuracy in time. On the basis of IMEX‐BDF2 time semi‐discrete method, we derive a fourth‐order compact (FOC) finite difference scheme for spatial discretization. Since the payoff function of the option at the strike price is not differentiable, the results show only second‐order accuracy in space. To remedy this, a local mesh refinement strategy is used near the strike price so that the accuracy achieves fourth order. Numerical results illustrate the effectiveness of the proposed method for European and American options under regime‐switching jump‐diffusion models.     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.     

Monotone Finite Difference Schemes for Nonlinear Systems with Mixed Quasi-Monotonicity

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.     

Generalized three-point difference schemes of high order of accuracy for nonlinear ordinary differential equations of the second order

For nonlinear ordinary differential equations of the second order with a derivative on the right-hand side and boundary conditions of the first kind, we construct and justify generalized three-point difference schemes of high order of accuracy on a nonuniform mesh. The existence and uniqueness of their solutions are proved, and an a priori estimate of the accuracy is obtained.     

An error analysis and the mesh independence principle for a nonlinear collocation problem

A nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton's method are studied. The mesh independence principle for the collocation problem is proved and used to develop an efficient multilevel solution method. Simple techniques are applied for estimating certain discretization and iteration constants that are used in the formulation of a mesh refinement strategy and an efficient multilevel method. Several mesh refinement strategies for solving a test problem are compared numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

拟线性椭圆边值问题有限元校正方法

§1.主要结果 考虑拟线性椭圆边值问题 u=0,在?Ω上,其中Ω是R~2中凸多边形,z=(x,y),D_1=?/?x,D_2=?/?y,a_i(z,ξ_0,ξ_1,ξ_2)是定义在Ω×R×R×R上的函数,适当光滑,i=0,1,2.定义