Math 712-001 Numerical Methods II

Introduction

• This is the sequel to math 614 Numerical Methods I. The course will cover basic aspects of various numerical techniques (derivation, analysis, and implementation) for solving initial value problems, boundary value problems, initial-boundary value problems of partial differential equations, including finite difference method, finite element method, spectral method, and boundary integral equation method.

Prerequisites:

• Numerical Methods I, PDE, and Programming Experience

References:

1. Lloyd N. Trefethen, " Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations " .
2. Arieh Iserles, "A First Course in the Numerical Analysis of Differential Equations".
3. J. W. Thomas, "Numerical Partial Differential Equations - Finite Difference Methods".
4. John C. Strikwerda, "Finite Difference Schemes and Partial Differential Equations".
5. K. W. Morton and D. F. Mayers, "Numerical Solution of Partial Differential Equations".
6. Robert D. Richtmyer and K. W. Morton, "Difference Methods for Initial-Value Problems".

Suggested Readings:

• Bengt Fornberg, " Generation of Finite Difference Formulas on Arbitrarily Spaced Grids ".

• SIAM News, Top Ten Algorithms of the 20th Century

• Jonathan Shewchuk, " An Introduction to the Conjugate Gradient Method Without the Agonizing Pain ".

• Rick Beatson and Leslie Greengard, " A Short Course on Fast Multipole Methods ".

• J.W. Cooley and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comp., 19 (1965), pp. 297–301.

• R. Courant, K. Friedrichs, and H. Lewy, "On the partial difference equations of mathematical physics," IBM J. Res. Develop., 11 (1967), pp. 215–234.

• R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations," Bull. Amer. Math. Soc., 49 (1943), pp. 1–23.

• A. Brandt, "Multi-level adaptive solutions to boundary-value problems," Math. Comp., 31 (1977), no. 138, pp. 333–390.

• M.R. Hestenes and E. Stiefel, "Methods of conjugate gradients for solving linear systems," J. Research Nat. Bur. Standards, 49 (1952), pp. 409–436.

• L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.

Web Resources

1. Codes

   • Netlib
   • GAMS (Guide to Available Mathematical Software)



2. Books

   • Abramowitz and Stegun, " Handbook of Mathematical Functions " (Bible).
   • Numerical Recipes in Fortran 77.
   • Yousef Saad, " Iterative methods for sparse linear systems ".

Homework Assignments