CATALOG DATA:
Interpolation, differentiation, integration, initial value and boundary value problems for ordinary differential equations, solution methods for parabolic, hyperbolic and elliptic partial differential equations; applications to classical and current research problems in engineering and sciences.

TEXTBOOK:
Press, Flannery, Teukolsky, and Vetterling, Numerical Recipes, The Art of Scientific Computing
Iserles, A., A First Course in the Numerical Analysis of Differential Equations

GOALS:
Be able to construct, analyze, and implement appropriate numerical methods for the solution of non-trivial ordinary and partial differential equations.

Topics:

• Interpolation: General interpolation problem, Lagrange polynomials, piecewise Lagrange interpolation, splines, parametric interpolation, multidimensional interpolation.
• Numerical differentiation: Finite differences form interpolation, finite differences from Taylor series, matrix representation of finite difference schemes, Hermitian methods and Padu approximations.
• Numerical integration: Newton-Cotes formula, 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 problem.
• Numerical solution of partial differential equations: 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 methods, direct methods, method of weighted residuals, Galerkin method, finite element method.

Computer Usage:
Extensive use of computers is made in each of the four homework sets (each consisting of 4-5 problems).

ABET category content as estimated by faculty member who prepared the course description:
Engineering Science: 3.0 credits or 100%
Engineering Design: 0 credits

Prepared by: Professor Samuel Paolucci
Last Update: March 10, 1997

Direct comments, questions, and corrections to amedept@nd.edu