Numerical Methods for PDEs Integral Equation Methods, Lecture 2 Numerical Quadrature Notes by Suvranu De and J. White Apl28,2003
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 SMA-HPC(2002 MIT Numerical Quadrature 0
3D Laplace's Basis Function Approach Equation Centroid Collocation Put collocation points at panel centroids dS nelj‖1x2=x A Collocation x A An a an」|乎 SMA-HPC(2002 MIT Numerical Quadrature 1
3D Laplace's Basis Function Approach Equation Calculating Matrix Elements ← Collocation oint panel j Panel j One point net area qu frature A proximation centroi Four point 0.25* Area quadrature A,≈∑ Approximation SMA-HPC(2002 MIT Numerical Quadrature 2
Normalized 1D Basis Function Approach Problem Collocation Discretization of 1D Equation (ac)= g(a, ac)o(a')dS 3 E[0, 1 Centroid collocated piecewise constant scheme xo=0 x x2 亚( c2/ g(Ec, n)dS to be evaluated SMA-HPC(2002 MIT Numerical Quadrature 3
Normalized 1D Simple Quadrature Scheme Problem f(e)da a f 2 Area under the curve Is approximated by a rectangle SMA-HPC(2002 MIT Numerical Quadrature 4
Normalized 1D Simple Quadrature Scheme Problem Improving the Accuracy (ak2(1) 1/3 Area under the curve Is approximated by two rectangles x SMA-HPC(2002 MIT Numerical Quadrature 5
Normalized 1D Simple Quadrature Scheme Problem General n-Point Formula 1/元 f(a)daec∑ Key questions about the method How fast do the errors decay with n? Are there better methods? SMA-HPC⊙2002MT Numerical Quadrature 6
Normalized 1D Simple Quadrature Scheme Problem Numerical Example sin()ac 80m 10 10 SMA-HPC(2002 MIT Numerical Quadrature 7
Normalized 1D General Quadrature Scheme Problem General 1D Form f(a)dae-∑ f(m2) 0 i=1 weight Evaluation Point Free to pick the evaluation points Free to pick the weights for each poin An n-point formula has 2n degrees of freedom! SMA-HPC(2002 MIT Numerical Quadrature 8