Numerical Analysis II

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Introduction to the numerical solution of initial and boundary value problems in differential equations.

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Topic Outline: 
  • Numerical methods for initial value problems Euler methods One-step and multi-step methods Consistency, stability, accuracy, accumulation of error, stiffness Stepsize selection and estimation of the error
  • Background material in two-point boundary value problems of ordinary differential equations: existence for linear problems, stability and regularity of solutions. Numerical methods for two point boundary value problems Discussion of boundary conditions One-step schemes, collocation Consistency, stability, error, and convergence Equi-distribution of error and mesh selection
  • Numerical solution of the heat equation Method of lines Implicit and explicit methods and the C.F.L.\ condition Consistency, stability, error and convergence
  • Implementation issues Advanced iterative methods for linear systems Using structure in linear systems Advanced issues in numerical solution of nonlinear systems
  • Numerical solution of eigenvalue problems Approximation of the principle eigenvalue Approximation of all eigenvalues Approximation of eigenfunctions
  • Advanced methods for solving matrix eigenvalue problems Householder matrices and orthogonal transformations, reduction of symmetric matrices to tridiagonal form