On the construction of a high-precision numerical calculation method for time-evolving partial differential equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03354
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Meiji Gakuin University (2023); Hachinohe Institute of Technology (2021-2022)
• Principal Investigator: TSUCHIYA Takuya 明治学院大学, 経済学部, 准教授 (50632139)
• Co-Investigator(Kenkyū-buntansha): 中村 誠 大阪大学, 大学院情報科学研究科, 教授 (70312634)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: 構造保存型数値解法 / 双曲型偏微分方程式 / 完全流体 / 曲がった時空

Outline of Final Research Achievements

We studied high-accuracy numerical methods for the Klein-Gordon equation and the perfect fluids in the curved spacetime. Using a structure-preserving numerical method, we derived some difference equations for the Klein-Gordon equation in curved spacetime and study their effects on the wave-forms of the solutions. We then clarified the cause of the numerical oscillations that occur from one of the difference equations. We also obtained the influence of space-time expansion on the solution. On the other hand, for perfect fluids, we performed some highly accurate simulations by coupling with the gravitational field equation.
Furthermore, we proposed a more stable system of gravitational field equation and analyzed the nonlinear Schrodinger equation in the de Sitter spacetime and the time evolved differential equation in the homogeneous and isotropic spacetime.

Free Research Field

数値解析

Academic Significance and Societal Importance of the Research Achievements

双曲型偏微分方程式においては、Laplace作用素を含む項が含まれ、この項から生じる離散化誤差が数値計算上の主要誤差になることが多い。そのため、Laplace作用素に対する離散化手法とその数値解に与える影響を調べることは、この項を含むすべての偏微分方程式の数値解析の発展に寄与すると考えられる。また、曲がった時空は偏微分作用素へ影響を与えるため、前述のLaplace作用素を含め偏微分方程式の安定性に寄与する可能性が高く、（偏）微分方程式の安定解析という分野に新たなアプローチが可能となると思われ、学術的意義が高いと考えらえる。