代用電荷法によるポアソン方程式の数値計算法の改良 An Improved Numerical Method for Poisson's Equation by the Charge Simulation Method

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著者

    • 森下 博 MORISHITA Hiroshi
    • 兵庫大学経済情報学部経済情報学科 Department of Economics and Imformation Science, Faculty of Economics and Information Science, Hyogo University
    • 天野 要 AMANO Kaname
    • 愛媛大学工学部情報工学科 Department of Computer Science, Faculty of Engineering, Ehime University
    • 四ツ谷晶二 YOTSUTANI Shoji
    • 龍谷大学理工学部数理情報学科 Department of Applied Mathematics and Infomatics, Faculty of Science and Technology, Ryukoku University

抄録

代用電荷法はラプラス方程式の高速で高精度な近似解法として知られている. しかし 調和関数を基本解の1次結合で近似するという方法の原理から ポアソン方程式には適していないと考えられてきた. 最近 この代用電荷法の利点を生かしたポアソン方程式の数値計算法が提案された. その方法は ポアソン方程式の解を特解と調和関数の和に分解し 前者を基本解で表現して数値積分し 後者を代用電荷法で近似する というものである. その有効性は数値実験的にも検証されている. しかし 同時に この方法には特解の数値積分に要する計算量が少なくないという大きな問題が指摘されていた. 特解の数値積分はポアソン方程式の解の精度を決める鍵ともなっている. 本稿では この特解の数値積分の問題に注目して 改良されたポアソン方程式の数値計算法を提案し その有効性を数値実験的に検証する. 具体的には 特解の数値積分法として (1)極座標を導入して対数ポテンシャルがら生じる特異性を取り除き (2)積分領域を境界値がなめらかな範囲で問題の領域を含む円領域に拡張し さらに (3)数値積分公式として 偏角方向には台形公式を 絶対値方向には二重指数関数型数値積分公式 (DE公式) を採用する. 数値実験の結果 実際に非常に精度の高い特解を得ることができる. 最終的なポアソン方程式の数値解の精度も大幅に向上し 計算時間も短縮される. ここで注意すべきは 積分領域の拡張によって台形公式・DE公式という積分公式の優れた特徴が十分に発揮されるということである. この方法の基本的な考え方は3次元問題にも適用可能である.The charge simulation method is known as a rapid and accurate solver for Laplace's equation, in which the solution is approximated by a linear combination of logarithmic potentials. It has been regarded as being unsuitable for Poisson's equation. However, a feasible method was recently presented for Poisson's equation making the best use of the charge simulation method. A particular solution is first obtained by numerical integration of the logarithmic formula. The problem is now reduced to Laplace's equation, which is approximated by the conventional charge simulation method. But, the method requires too much computation in the numerical integration of the particular solution, which is also a key to the accuracy of final results. In this paper, we propose an improved numerical method for solving the Dirichlet problem of Poisson's equation paying special attention to the numerical integration of the particular solution. We (1) remove the singurality caused by logarithmic potential by using the polar coordinate system, (2) extend the integral domain to a disk including the original problem domain, and (3) apply Trapezoidal formula and Double Exponential formula to the numerical integration. The combination of (1)縲鰀(3) results in high accuracy of the particular solution and the final results, and also in a reduction of the computational cost. The basic idea of the method is applicable also to three dimensional problems.

The charge simulation method is known as a rapid and accurate solver for Laplace's equation, in which the solution is approximated by a linear combination of logarithmic potentials. It has been regarded as being unsuitable for Poisson's equation. However, a feasible method was recently presented for Poisson's equation making the best use of the charge simulation method. A particular solution is first obtained by numerical integration of the logarithmic formula. The problem is now reduced to Laplace's equation, which is approximated by the conventional charge simulation method. But, the method requires too much computation in the numerical integration of the particular solution, which is also a key to the accuracy of final results. In this paper, we propose an improved numerical method for solving the Dirichlet problem of Poisson's equation paying special attention to the numerical integration of the particular solution. We (1) remove the singurality caused by logarithmic potential by using the polar coordinate system, (2) extend the integral domain to a disk including the original problem domain, and (3) apply Trapezoidal formula and Double Exponential formula to the numerical integration. The combination of (1)〜(3) results in high accuracy of the particular solution and the final results, and also in a reduction of the computational cost. The basic idea of the method is applicable also to three dimensional problems.

収録刊行物

  • 情報処理学会論文誌

    情報処理学会論文誌 40(9), 3337-3344, 1999-09-15

    一般社団法人情報処理学会

参考文献:  15件中 1-15件 を表示

各種コード

  • NII論文ID(NAID)
    110002725056
  • NII書誌ID(NCID)
    AN00116647
  • 本文言語コード
    JPN
  • 資料種別
    Journal Article
  • ISSN
    1882-7764
  • NDL 記事登録ID
    4851358
  • NDL 雑誌分類
    ZM13(科学技術--科学技術一般--データ処理・計算機)
  • NDL 請求記号
    Z14-741
  • データ提供元
    CJP書誌  NDL  NII-ELS  IPSJ 
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