• 论文 •

### 非线性Leland方程的一种并行本性差分方法

1. 华北电力大学数理学院, 北京 102206
• 收稿日期:2013-08-02 出版日期:2014-03-15 发布日期:2014-03-14
• 基金资助:

国家自然科学基金（11371135，10771065），中央高校基本科研业务 费专项资金资助（13QN30）和 北京市共建项目专项资助（2012年）

Wu Lifei, Yang Xiaozhong, Zhang Fan. A KIND OF DIFFERENCE METHOD WITH INTRINSIC PARALLELISM FOR NONLINEAR LELAND EQUATION[J]. Journal of Numerical Methods and Computer Applications, 2014, 35(1): 69-80.

### A KIND OF DIFFERENCE METHOD WITH INTRINSIC PARALLELISM FOR NONLINEAR LELAND EQUATION

Wu Lifei, Yang Xiaozhong, Zhang Fan

1. Mathematics and Physics Department, North China Electric Power University, Beijing 102206, China
• Received:2013-08-02 Online:2014-03-15 Published:2014-03-14

It is very important to study the numerical solution of nonlinear Leland equation (option pricing model with transaction costs). For solving nonlinear Leland equation, a difference method with intrinsic parallelism–Alternating Segment Crank-Nicolson (ASC-N) scheme is constructed in this paper. Then the existence and uniqueness, computational stability and error estimate of ASC-N scheme are analyzed. Theoretical analysis demonstrates that ASC-N scheme is unconditional stability parallel difference scheme. Numerical experiment demonstrates that computational accuracy of ASC-N scheme is closed to classics Crank-Nicolson scheme. But the computational time of ASC-N scheme can save nearly 50% for classics Crank-Nicolson scheme. Theoretical analysis and numerical experiment demonstrate the superiority of ASC-N scheme for solving nonlinear Leland equation.

MR(2010)主题分类:

()
 [1] Kwok Y K. Mathematical models of financial derivatives (2nd Edition)[M]. Beijing: The World Book Publishing Company, 2011.[2] 姜礼尚. 期权定价的数学模型和方法(第二版)[M]. 北京: 高等 教育出版社, 2008.[3] Julia A, Matthias E. On the numerical solution of nonlinear Black-Scholes equations[J]. Computers and Mathematics with Applications, 2008, 56(3): 799-812.[4] Company R, Navarro E, Pintos J R, et al. Numerical solution of linear and nonlinear Black-Scholes option pricing equations[J]. Computers and Mathematics with Applications, 2008, 56(3): 813-821.[5] Yang X Z, Liu Y G, Wang G H. A study on a new kind of universal difference schemes for solving Black-Scholes equation[J]. International Journal of Information and Science System, 2007, 3(2): 251-260.[6] 吴立飞, 杨晓忠. 支付红利下Black-Scholes方程的显-隐和隐-显差 分格式解法[J]. 中国科技论文在线精品论文, 2011, 4(13): 1207-1212.[7] D. J.Evans, A.R.B.Abdullah. Group explicit methods for parabolic equations[J]. Journal Computer Math., 1983, 14: 145-154.[8] Zhang B L. Alternating segment Explicit-Implicit Methods for Diffusion Equation. J. Numer. Method Comput. Appl., 1991, 4: 245-253.[9] 张宝琳 等. 偏微分方程并行有限差分方法[M]. 北京: 科学出版 社, 1994.[10] 王文洽. KdV方程的一类本性并行差分格式[J]. 应用数学学报, 2006, 29(6): 995-1003.[11] Sheng Z Q, Yuan G W, Hang X D. Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation[J]. Appl. Math. Comput., 2007, 184: 1015-1031.[12] Yuan G W, Sheng Z Q. The unconditional stability of parallel difference with second order convergence for nonlinear parabolic system[J]. J. Paratial Difference Equations, 2007, 20: 45-64.[13] 袁光伟, 岳晶岩, 盛志强等. 非线性抛物型方程计算方法[J]. 中国科学:数学, 2013, 43(3): 235-248.[14] 刘维. 实战Matlab之并行程序设计[M]. 北京: 北京航空航天大学 出版社, 2012.
 [1] 何鼎, 胡鹏. 求解随机常微分方程的平均单支$\theta-$方法[J]. 数值计算与计算机应用, 2022, 43(1): 49-60. [2] 杜玉龙, 徐凯文, 赵昆磊, 袁礼. 基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器[J]. 数值计算与计算机应用, 2021, 42(2): 155-168. [3] 邱泽山, 曹学年. 带漂移的单侧正规化回火分数阶扩散方程的三阶数值格式[J]. 数值计算与计算机应用, 2020, 41(3): 201-215. [4] 刘新龙, 杨晓忠. 时间分数阶四阶扩散方程的显-隐和隐-显差分格式[J]. 数值计算与计算机应用, 2020, 41(3): 216-231. [5] 关文绘, 曹学年. Riesz回火分数阶平流-扩散方程的隐式中点方法[J]. 数值计算与计算机应用, 2020, 41(1): 42-57. [6] 洪旗, 苏帅. 任意四边形网格上扩散问题的一个稳定九点格式[J]. 数值计算与计算机应用, 2019, 40(1): 51-67. [7] 丛玉豪, 赵欢欢, 张艳. 中立型时滞微分系统多步龙格-库塔方法的时滞相关稳定性[J]. 数值计算与计算机应用, 2018, 39(4): 310-320. [8] 毛文亭, 张维, 王文强. 一类带乘性噪声随机分数阶微分方程数值方法的弱收敛性与弱稳定性[J]. 数值计算与计算机应用, 2018, 39(3): 161-171. [9] 邓维山, 徐进. 一种泊松-玻尔兹曼方程稳定算法的高效有限元并行实现[J]. 数值计算与计算机应用, 2018, 39(2): 91-110. [10] 吴长茂, 杨超, 尹亮, 刘芳芳, 孙乔, 李力刚. 基于CPU-MIC异构众核环境的行星流体动力学数值模拟[J]. 数值计算与计算机应用, 2017, 38(3): 197-214. [11] 宋梦召, 冯仰德, 聂宁明, 王武. 基于空位跃迁的KMC并行实现[J]. 数值计算与计算机应用, 2017, 38(2): 130-142. [12] 刘辉, 冷伟, 崔涛. 并行自适应有限元计算中的负载平衡研究[J]. 数值计算与计算机应用, 2015, 36(3): 166-184. [13] 刘辉, 崔涛, 冷伟. hp自适应有限元计算中一种新的自适应策略[J]. 数值计算与计算机应用, 2015, 36(2): 100-112. [14] 刘翠翠, 张瑞平. 基于勒让德多项式逼近的4级4阶隐式Runge-Kutta方法[J]. 数值计算与计算机应用, 2015, 36(1): 22-30. [15] 王文强, 孙晓莉. 一类随机分数阶微分方程隐式Euler方法的弱收敛性与弱稳定性[J]. 数值计算与计算机应用, 2014, 35(2): 153-162.