Geometric structure-preserving finite difference method for vector-valued evolution equation

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654026
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Shibaura Institute of Technology
• Principal Investigator: ISHIWATA Tetsuya 芝浦工業大学, システム工学部, 准教授 (50334917)
• Co-Investigator(Kenkyū-buntansha): FURIHATA Daisuke 大阪大学, サイバーメティアセンター, 准教授 (80242014)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 数値解析 / 差分法 / 構造保存 / ベクトル値関数 / 非線形偏微分方程式 / 幾何学的構造

Outline of Final Research Achievements

We propose effective numerical schemes for vector-valued partial differential equations in engineering fields and give some theoretical analyses to
these schemes. The target problem are the Landau-Lifshitz equation, Ericksen-Leslile equation, Localized Induction model and so on. We construct structure-preserving schemes for these models, that is, the proposed schemes inherit a geometric property and energy structure from the original problems and show the existence and uniqueness of the finite difference solutions and also error estimates. For the problem which exact solutions are known or constructible, we compare the numerical solutions with the exact solutions and show the effectiveness of our proposed schemes.

Free Research Field

応用数学