Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

详细信息    查看全文

• 作者：Avram Sidi (1)
  
• 关键词：Cauchy principal value ; Hadamard finite part ; Singular integral ; Hypersingular integral ; Numerical quadrature ; Trapezoidal rule ; Euler鈥揗aclaurin expansion ; Richardson extrapolation ; 41A55 ; 41A60 ; 45B05 ; 45B15 ; 45E05 ; 45E99 ; 65B05 ; 65B15 ; 65D30 ; 65D32
• 刊名：Journal of Scientific Computing
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：60
• 期：1
• 页码：141-159
• 全文大小：
• 参考文献：1. Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)
  2. Chan, Y.S., Fannjiang, A.C., Paulino, G.H.: Integral equations with hypersingular kernels鈥攖heory and applications to fracture mechanics. Int. J. Eng. Sci. 41, 683鈥?20 (2003) CrossRef
  3. Chen, J.T., Kuo, S.R., Lin, J.H.: Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity. Int. J. Numer. Methods Eng. 54, 1669鈥?681 (2002) CrossRef
  4. Chen, Y.Z.: Hypersingular integral equation method for three-dimensional crack problem in shear mode. Commun. Numer. Methods Eng. 20, 441鈥?54 (2004) CrossRef
  5. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
  6. Evans, G.: Practical Numerical Integration. Wiley, New York (1993)
  7. Kress, R.: On the numerical solution of a hypersingular integral equation in scattering theory. J. Comput. Appl. Math. 61, 345鈥?60 (1995) CrossRef
  8. Kress, R., Lee, K.M.: Integral equation methods for scattering from an impedance crack. J. Comput. Appl. Math. 161, 161鈥?77 (2003) CrossRef
  9. Kythe, P.K., Sch盲ferkotter, M.R.: Handbook of Computational Methods for Integration. Chapman & Hall/CRC Press, New York (2005)
  10. Ladopoulos, E.G.: Singular Integral Equations: Linear and Non-Linear Theory and its Applications in Science and Engineering. Springer, Berlin (2000) CrossRef
  11. Lifanov, I.K., Poltavskii, L.N., Vainikko, G.M.: Hypersingular Integral Equations and Their Applications. CRC Press, New York (2004)
  12. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
  13. Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Number 10 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003) CrossRef
  14. Sidi, A.: Euler鈥揗aclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math. 98, 371鈥?87 (2004) CrossRef
  15. Sidi, A.: Euler鈥揗aclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comput. 81, 2159鈥?173 (2012) CrossRef
  16. Sidi, A.: Euler鈥揗aclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36, 331鈥?52 (2012)
  17. Sidi, A.: Compact numerical quadrature formulas for hypersingular integrals and integral equations. J. Sci. Comput. 54, 145鈥?76 (2013) CrossRef
  18. Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3, 201鈥?31 (1988). Originally appeared as technical report no. 384, Computer Science Department, Technion-Israel Institute of Technology (1985), and also as ICASE report no. 86-50 (1986)
  19. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 3rd edn. Springer, New York (2002) CrossRef
  
• 作者单位：Avram Sidi (1)
  
  1. Computer Science Department, Technion-Israel Institute of Technology, Haifa, 32000, Israel
  
• ISSN：1573-7691

文摘

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals \(I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx\) and \(I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx\) . These integrals are not defined in the regular sense; \(I^{(1)}[g]\) is defined in the sense of Cauchy Principal Value while \(I^{(2)}[g]\) is defined in the sense of Hadamard Finite Part. With \(h=(b-a)/n, \,n=1,2,\ldots \) , and \(t=a+kh\) for some \(k\in \{1,\ldots ,n-1\}, \,t\) being fixed, the numerical quadrature formulas \({Q}^{(1)}_n[g]\) for \(I^{(1)}[g]\) and \(Q^{(2)}_n[g]\) for \(I^{(2)}[g]\) are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that \(g\in C^\infty [a,b]\) . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants \(c^{(k)}_i\) being independent of \(h\) . In this work, we apply the Richardson extrapolation to \({Q}^{(k)}_n[g]\) to obtain approximations of very high accuracy to \(I^{(k)}[g]\) . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function \(g(x)\) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.