Numerical methods for solving partial differential equations on surfaces

求解曲面上偏微分方程的数值方法

• 批准号: 227823-2009
• 负责人: Ruuth, Steven
• 金额: $ 1.6万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=425286
• 关键词: Numerical; methods; solving; partial; differential

Partial differential equations (PDEs) are ubiquitous throughout the sciences and applied sciences. Because analytical solutions are rarely possible, the practical importance of accurate and efficient numerical methods for PDEs cannot be overemphasized. There has been a great effort made to develop numerical methods for many important classes of PDEs. These efforts focus on finding solutions in one, two or three spatial dimensions. But problems involving general differential equations also arise on general manifolds, such as two-dimensional curved surfaces or one-dimensional curved filaments. For example, in material science one might wish to examine phase change of a material on a curved surface. In biological modeling, one might study wound healing on skin, or the evolution of a pattern on an animal coat. Computer graphics and image processing also frequently use PDEs on surfaces; for example, they might use such methods to place a texture on a surface or restore a damaged pattern on a surface such as a vase. Despite the widespread occurrence of PDEs on curved surfaces there is still a need for a systematic approach for accurately computing the solution of general PDEs on general surfaces.

偏微分方程（PDE）在科学和应用科学中无处不在。由于解析解很少是可能的，准确和有效的数值方法的偏微分方程的实际重要性，不能过分强调。已经有了很大的努力，发展的许多重要类的偏微分方程的数值方法。这些努力侧重于在一个、两个或三个空间维度上找到解决办法。但涉及一般微分方程的问题也出现在一般流形上，如二维曲面或一维弯曲细丝。 例如，在材料科学中，人们可能希望检查曲面上材料的相变。在生物建模中，人们可以研究皮肤上的伤口愈合，或者动物皮毛上图案的进化。计算机图形学和图像处理也经常在表面上使用偏微分方程;例如，它们可能会使用这种方法在表面上放置纹理或恢复花瓶等表面上的损坏图案。尽管在曲面上的偏微分方程的广泛发生，仍然需要一个系统的方法来精确地计算一般曲面上的一般偏微分方程的解决方案。