Structure-preserving numerical method for partial differential equations based on Voronoi diagram

2016 Fiscal Year Final Research Report

• Project/Area Number: 26610038
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Osaka University
• Principal Investigator: Furihata Daisuke 大阪大学, サイバーメディアセンター, 准教授 (80242014)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 構造保存数値解法 / 離散変分 / ボロノイ分割 / 非構造格子

Outline of Final Research Achievements

The discretization of the differential limiting operation is the base of the original finite difference method. We appropriately introduced it to our study via the Voronoi decomposition and derived the exact discrete Green theorem by Voronoi discretization on the arbitrary lattice in the multidimensional region, e.g., the gradient, Laplacian.Our results include mathematical proofs and mathematically rigorous.
This showed that the Voronoi-Delaunay triangulation has favorable characteristics to allow arbitrary lattice placement while inheriting excellent mathematical properties.Furthermore, we adopted this Voronoi difference as the basis of the discrete variational derivative method to design numerical schemes for some partial differential equations.This indicates that our studies strengthened both the practicality and the theoretical foundation of the numerical solution method of partial differential equations.

Free Research Field

応用数学