徐俊林
求偏微分方程的数值解(有限差分法、配置法、Galerkin方法)
2021-6-22 12:09
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偏微分方程PDE通常用于描述各种各样的现象,例如声、热、电、电动力学、弹性力学、流体力学等。

计算环境:Maple 2015 - Maple 2021

计算文件下载:

pde_approximation.zip

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Finite difference method

  • approximate differential equations using finite difference equations to approximate derivatives


Collocation method

  • uses a finite-dimensional space of basis functions and collocation points to approximate PDEs


Galerkin's technique

  • uses orthogonality of a set of basis function to turn PDEs into coupled sets of ODEs


Finite difference method

  • The derivatives of the PDE are approximated by linear combinations of function values at the structured grid points, using a Taylor series expansion

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Collocation method

  • Define the basis function (usually polynomials) and approximate solution:

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  • Calculate residual function by substituting the candidate function into the original PDE

  • Choose the collocation points at which the candidate function must exactly match


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Galerkin's method

  • Define the basis function (usually orthogonal polynomials) and approximate solution as

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  • Calculate residual function by substituting the candidate function into the original PDE

  • Minimize the residual using Least Squares Error 

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Maple中的计算文件:

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基于PDE的应用示例:

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Whitepaper-Developing_Mathematical_Models_of_Batteries.pdf


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