Discrete integration by parts on any convex polygon and design of structure-preserving numerical schemes

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K20883
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Osaka University
• Principal Investigator: Furihata Daisuke 大阪大学, サイバーメディアセンター, 教授 (80242014)
• Project Period (FY): 2020-07-30 – 2024-03-31
• Keywords: 離散部分積分公式 / 構造保存数値解法 / 差分法 / 対数差分

Outline of Final Research Achievements

Based on the discretization of base space and differential operators, we constructed a discrete vector analysis for a piecewise constant function space on discrete convex polygons. We can discretely reproduce some primary laws of vector analysis and prove their properties. We also can design structure-preserving numerical schemes and verified this through numerical experiments.
We also investigated that applying fasten methods for existing structure-preserving numerical solutions to the new discretization method described above is possible. Our research has led to the construction of a new difference operator, a significant advancement in our field. This operator is spatially symmetric, ensuring high numerical stability, and allows for precise control of the error profile. We were able to introduce excellent properties as a nonlinear function of the function value on the reference point, a feature that has practical implications for error control in numerical analysis.

Free Research Field

数値解析学

Academic Significance and Societal Importance of the Research Achievements

構造保存数値解法とは微分方程式がもつ数学的性質を保存する数値解法であり，複雑さや非線形性の強い問題，超長期軌道計算が必要な問題等の分野では大変重要な数値解法である．しかし定義領域離散化手法が限定的であった．これは任意格子上での離散変分計算を行うことができなかった数学的な事情による．この状況に対しわれわれは自然な数学的拡張により任意凸多角形格子上での離散変分計算を可能とする，この困難を克服する突破口を見出した．自由格子上での数値計算は理想的だが，これまでは同時に数学的性質の多くを失うものであった．これに対し本研究はこの困難を克服し，新しい方向性を創り出せると期待したものである．