Numerical Methods for PDEs, Integral Equation Methods, Lecture 2: Numerical Quadrature - Edubirdie

University:
Massachusetts Institute of Technology
Course:
16.920J | Numerical Methods for Partial Differential Equations (SMA 5212)
Academic year:

2023
Views:

438

Pages:

21
Author:

Jacoby Robles

1 Outline Easy technique for computing integrals Piecewise constant approach. Gaussian Quadrature Convergence properties Essential role of orthogonal polynomials Multidimensional Integrals Techniques for singular kernels Adaptation and variable transformation Singular quadrature. 2 3D Laplace's Equation 2.1 Basis Function Approach 2.1.1 Centroid Collocation Put collocation points at panel centroids A Note 1 In the last lecture we were introduced to integral equations and several different techniques for discretizing them were described. It was pointed out that one of the most popular means of obtaining a discrete set of equations is to use a piecewise constant centroid collocation scheme. We consider a simple problem of solvitug Laplace's equation in 3D. The potential, is prescribed on the sur- face of the cube and we need to compute the charge distribution o. In order to do that we break the surface of the cube up into n panels and assume a constant charge distribution on each panel. Mathematically, this corresponds to assuming piecewise constant basis functions, each basis function, being compactly supported on the it panel. The resulting semi-discrete equation is a function of the spatial variable z. In order to obtain a discrete set of equations, we assume that this semi-discrete equation is satisfied exactly at the centroids of the panels. This gives rise to the set of n equations corresponding to the n panels. Mathematically, this process of collocation corresponds to setting the residual orthogonal to a set of delta functions located at the panel centroids. 1 3.3.8 Computing the Points and Weights Could use Newton's Method. F(y)=0J (y) (y+1)=F() The nonlinear function for Newton is then 0 xdx Note 13 Newton's method is an iterative technique for finding a value y such that F(y)= 0. The method is based on linearizing the problem about a guess at y, and then updating the value of jy by solving the linearized problem. In particular, the iterate y is determined from y by solving the linear system of equations F() + Jr() (+1)=0 where Jp (y) is the Jacobian (multidimensional derivative) of the nonlinear function F(y). The iteration is continued until the updated y is sufficiently. close to the exact solution, a criterion that can be difficult to verify. Newton's method does not always converge, a phenomenon that is more likely when Jy(y) is nearly singular. For more about Newton's method, see the 6.336/16.920/2.006 course notes. 3.3.9 Computing the Points and Weights Newton Method Jacobian reveals problem 2n 00 0 Columns become Enearly dependent for high ander Note 14 Looking at the Jacobian of the problem, we realize that the first 'n' columns become increasingly linearly dependent for large 1. This is bound to happen since 11

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