Catalog description

• The course is directed at first year graduate students in engineering and science. It is open to undergraduates who wish to become better prepared for graduate studies or become more proficient in the techniques of numerical analysis.

Offerings by Powers

• Next: ?
• Present: Fall 2022
• Previous: Fall 2012, Fall 2010, Fall 2007, Fall 2004, Fall 2003, Fall 1999

Prerequisites

• Formally none; undergraduate course work in linear algebra, ordinary and partial differential equations, and computer programming will be very important

Topics

• Interpolation
General interpolation problem; Lagrange polynomials; piecewise Lagrange interpolation; splines; parametric interpolation; multidimensional interpolation.
• Numerical Differentiation
Finite differences from interpolation; finite differences from Taylor series; matrix representation of finite difference schemes; Hermitian methods and Pade approximations.
• Numerical Integration
Newton-Cotes formulas; trapezoidal rule; Simpson's rule; error analysis; trapezoidal rule with end correction; Richardson's extrapolation and Romberg integration; adaptive quadrature; Gauss quadrature; semi-infinite intervals; infinite intervals; singularities.
• Numerical Solution of Ordinary Differential Equations
Initial value problems; Runge-Kutta type formulas; multi-step methods; implicit methods; accuracy; stability; model equation; phase and amplitude errors; system of differential equations; stiffness; boundary value problems; shooting; direct methods; non-uniform grids; eigenvalue problems.
• Numerical Solution of Partial Differential Equations
Method of lines for parabolic and hyperbolic PDEs; modified wave number and von Neumann stability analysis; modified equation analysis; alternating direction implicit methods; approximate factorization; nonlinear equations; iterative methods for elliptic PDEs; conjugate gradient methods; multigrid method; direct methods; method of weighted residuals; Galerkin method; pseudospectral method; finite element method.