Geometric Numerical Integration Methods for Differential-Algebraic Equations and Their Application to Evolutionary Equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K23399
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: Sato Shun 東京大学, 大学院情報理工学系研究科, 助教 (40849072)
• Project Period (FY): 2019-08-30 – 2023-03-31
• Keywords: 構造保存数値解法 / 常微分方程式 / 微分代数方程式 / 偏微分方程式 / 陰的線形スキーム

Outline of Final Research Achievements

Structure-preserving numerical methods have been well developed for ordinary differential equations (ODEs), but they have not yet been well studied for differential-algebraic equations (DAEs), which are a generalization of ODEs and frequently appear as models of systems with constraints. This study aims at developing structure-preserving numerical methods for ODEs and DAEs, and their application to partial differential equations.
Based on these objectives, we have proposed a gradient flow interpretation of the scalar auxiliary variable methods, constructed and analyzed high-order linearly implicit structure-preserving numerical methods for ODEs with quadratic invariants, and applied some structure-preserving numerical methods to a PDE with constraints.

Free Research Field

数値解析学

Academic Significance and Societal Importance of the Research Achievements

微分方程式の数値解法は現代科学のさまざまな分野において重要な役割を担っている．中でも，微分方程式の構造 (保存量や対称性など) を尊重した構造保存数値解法の有効性が20世紀末に認識され，今では広く利用されている．
本研究は構造保存数値解法の適用対象を広げるものであり，今後の数値シミュレーションにおいて有用であると期待される．