三维动边值问题的迎风差分方法

数学物理学报 ›› 2012, Vol. 32 ›› Issue (2): 271-289.

三维动边值问题的迎风差分方法

袁益让¹|李长峰²

1.山东大学数学研究所济南 250100|2.山东大学经济学院济南 250100

收稿日期:2010-10-08 修回日期:2011-08-06 出版日期:2012-04-25 发布日期:2012-04-25
基金资助:
国家重点基础研究专项经费项目(G19990328)、国家攻关项目(20050200069)、国家自然科学基金(10771124, 10372052, 11101244)、
国家教育部博士点基金(20030422047)、山东省自然科学基金(ZR2009AQ012)和山东大学自主创新基金(2010TS031)资助

The Upwind Finite Difference Method for Three-dimensional Moving Boundary Value Problem

YUAN Yi-Rang¹, LI Chang-Feng²

1.Institute of Mathematics, Shandong University, Jinan 250100;
2.School of Economics, Shandong University, Jinan |250100

Received:2010-10-08 Revised:2011-08-06 Online:2012-04-25 Published:2012-04-25
Supported by:
国家重点基础研究专项经费项目(G19990328)、国家攻关项目(20050200069)、国家自然科学基金(10771124, 10372052, 11101244)、
国家教育部博士点基金(20030422047)、山东省自然科学基金(ZR2009AQ012)和山东大学自主创新基金(2010TS031)资助

摘要/Abstract

摘要：

可压缩可混溶油、水三维渗流动边值问题的研究,对重建盆地发育中油气资源运移、聚集的历史和评估油气资源的勘探与开发有重要的价值,
其数学模型是一组非线性耦合偏微分方程组的动边值问题. 该文对有界域的动边值问题提出一类新的二阶修正迎风差分格式, 应用区域变换、
变分形式、能量方法、差分算子乘积交换性理论、高阶差分算子的分解、微分方程先验估计的理论和技巧, 得到了最佳 $l^2$ 误差估计结果.
该方法已成功应用到油资评估的数值模拟中. 它对这一领域的模型分析, 数值方法和软件研制均有重要的价值.

关键词: 可压缩渗流, 三维动边界, 迎风分数步差分, 最佳l² 估计, 实际应用

Abstract:

The research of the three-dimensional compressible miscible (oil and water) displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of three-dimensional bounded region, the authors put forward a kind of upwind finite difference schemes and make use the calculus of variations, the change of variables and the theory of a priori estimates and techniques. Optimal order estimates in l² norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, model numerical method and software development. Thus, the well-known problem is solved.

Key words: Compressible displacement, Three-dimensional moving boundary, Upwind finite difference fractional steps, l²error estimate, Actual application

中图分类号:

65M06

袁益让, 李长峰. 三维动边值问题的迎风差分方法[J]. 数学物理学报, 2012, 32(2): 271-289.

YUAN Yi-Rang, LI Chang-Feng. The Upwind Finite Difference Method for Three-dimensional Moving Boundary Value Problem[J]. Acta mathematica scientia,Series A, 2012, 32(2): 271-289.

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