Textbooks
The principal textbooks for 18.086 are:
Strang, Gilbert. Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press, 1986. ISBN: 9780961408800. (Table of Contents)
———. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.
Draft versions of chapters relevant to this course are available below.
Section 3.5 - Finite Differences and Fast Poisson Solvers (PDF)
Section 5.1- Finite Difference Methods (PDF)
Section 5.2 - Accuracy and Stability for $u_t= cu_x$ (PDF)
Section 5.3 - The Wave Equation and Staggered Leapfrog (PDF)
Section 5.4 - The Heat Equation and Convection-Diffusion (PDF)
Section 5.5 - Difference Matrices and Eigenvalues (PDF)
Section 5.6 - Nonlinear Flow and Conservation Laws (PDF)
Section 5.7 - Level Sets and the Fast Marching Method (PDF)
Section 6.1 - Elimination with Reordering (PDF)
Section 6.2 - Iterative Methods (PDF)
Section 6.3 - Multigrid Methods (PDF)
Section 6.4 - Krylov Subspaces and Conjugate Gradients (PDF)
Section 6.5 - The Saddle Point Stokes Problem (PDF)
Section 7.1 - One Fundamental Example (PDF)
Section 7.2 - Calculus of Variations (PDF)
Supplementary Readings
Anderson, E., et al. LAPACK User’s Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999.
Ascher, U. M., R. M. M. Mattheij, and R. D. Russell. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Upper Saddle River, NJ: Prentice Hall, 1988.
Axelsson, O. Iterative Solution Methods. Cambridge, UK: Cambridge University Press, 1994.
Axelsson, O., and V. A. Barker. Finite Element Solution of Boundary Value Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001.
Bathe, K. J. Finite Element Procedures. Upper Saddle River, NJ: Prentice Hall, 1996.
Bender, C. M., and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. New York, NY: McGraw-Hill, 1978.
Björck, A. Numerical Methods for Least Squares Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996.
Bracewell, R. N. The Fourier Transform and Its Applications. New York, NY: McGraw-Hill, 1986.
Brenner, S. C., and L. R. Scott. The Mathematical Theory of Finite Element Methods. New York, NY: Springer-Verlag, 1994.
Brezzi, F., and M. Fortin. Mixed and Hybrid Finite Element Methods. New York, NY: Springer-Verlag, 1991.
Briggs, W. L., V. Henson, and S. F. McCormick. A Multigrid Tutorial. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
Butcher, J. C. The Numerical Analysis of Ordinary Differential Equations. New York, NY: John Wiley & Sons, 1987.
Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics. New York, NY: Springer-Verlag, 1987.
Ciarlet, P. G. The Finite Element Method for Elliptic Problems. Amsterdam, NL: North-Holland Pub. Co., 1978.
Dongarra, J. J., I. S. Duff, D. C. Sorensen, and H. Van der Vorst. Numerical Linear Algebra for High-Performance Computers. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1998.
Elman, H., D. Silvester, and A. Wathen. Finite Elements and Fast Iterative Solvers. New York, NY: Oxford University Press, 2005.
Fornberg, B. A Practical Guide to Pseudospectral Methods. Cambridge, UK: Cambridge University Press, 1996.
Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations. Upper Saddle River, NJ: Prentice Hall, 1971.
George, A., and J. W. Liu. Computer Solution of Large Sparse Positive Definite Systems. Upper Saddle River, NJ: Prentice Hall, 1981.
Golub, G. H., M. Benzi, and M. J. Gander. “Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-point Problems.” BIT. Numerical Mathematics 43 (2003): 881-900.
Golub, G. H., and C. F. Van Loan. Matrix Computations. Baltimore, MD: Johns Hopkins University Press, 1996.
Greenbaum, A. Iterative Methods for Solving Linear Systems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997.
Gresho, P., and R. Sani. Incompressible Flow and the Finite Element Method. Volume 1: Advection-Diffusion. New York, NY: John Wiley & Sons, 1998.
Griffiths, D. F., and G. A. Watson, eds. Numeral Analysis. New York, NY: Longmans, 1986.
Gustafsson, B., H. -O. Kreiss, and J. Oliger. Time-Dependent Problems and Difference Methods. New York, NY: John Wiley & Sons, 1995.
Hackbusch, W. Multigrid Methods and Applications. New York, NY: Springer-Verlag, 1985.
Hackbusch, W., and U. Trottenberg. Multigrid Methods. Lecture Notes in Math. Vol. 960. New York, NY: Springer-Verlag, 1982.
Hairer, E., S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. New York, NY: Springer-Verlag, 1987.
Hairer, E., and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. New York, NY: Springer-Verlag, 1991.
Henrici, P. Discrete Variable Methods in Ordinary Differential Equations. New York, NY: John Wiley & Sons, 1962.
———. Applied and Computational Complex Analysis, III. New York, NY: John Wiley & Sons, 1986.
Higham, D. J., and N. J. Higham. MATLAB® Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2005.
Higham, N. J. Accuracy and Stability of Numeral Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996.
Hughes, T. J. R. The Finite Element Method. Upper Saddle River, NJ: Prentice Hall, 1987.
Iserles, A. A First Course in the Numerical Analysis of Differential Equations. Cambridge, UK: Cambridge University Press, 1996.
Johnson, C. Numerical Solutions of Partial Differential Equations by The Finite Element Method. Cambridge, UK: Cambridge University Press, 1987.
Keller, H. B. Numerical Solution of Two Point Boundary Value Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1976.
Kelley, C. T. Iterative Methods for Optimization. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999.
Lambert, J. D. Numerical Methods for Ordinary Differential Systems. New York, NY: John Wiley & Sons, 1991.
Lax, P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1973.
LeVeque, R. J. Numerical Methods for Conservation Laws. 2nd ed. Boston, MA: Birkhäuser Verlag, 1992.
Mallat, S. A Wavelet Tour of Signal Processing. Burlington, MA: Academic Press, 1999.
Oppenheim, A. V., and R. W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1989.
Parlett, B. N. The Symmetric Eigenvalue Problem. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1998.
Petzold, L. R. “A Description of DASSL - A Differential Algebraic System Solver.” IMACS Trans Sci Comp 1 (1982): 1-65. North Holland.
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes. Cambridge, UK: Cambridge University Press, 1986.
Quarteroni, A., and A. Valli. Numerical Approximation of Partial Differential Equations. New York, NY: Springer-Verlag, 1997.
Richtmyer, R. D., and K. W. Morton. Difference Methods for Initial-Value Problems. New York, NY: John Wiley & Sons, 1967.
Saad, Y. Iterative Methods for Sparse Linear Systems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2003.
Shampine, L. F., and C. W. Gear. “A User’s View of Solving Stiff Ordinary Differential Equations.” SIAM Review 21 (1979): 1-17.
Sod, G. A. Numerical Methods in Fluid Dynamics. Cambridge, UK: Cambridge University Press, 1985.
Stoer, J., and R. Bulirsch. Introduction to Numerical Analysis. New York, NY: Springer-Verlag, 2002.
Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 2003.
Strang, G., and K. Borre. Linear Algebra, Geodesy, and GPS. Wellesley, MA: Wellesley-Cambridge Press, 1997.
Strang, G., and G. J. Fix. An Analysis of the Finite Element Method. Upper Saddle River, NJ: Prentice Hall, 1973; Wellesley, MA: Wellesley-Cambridge Press, 2000.
Strang, G., and T. Nguyen. Wavelets and Filter Banks. Wellesley, MA: Wellesley-Cambridge Press, 1996.
Strikwerda, J. C. Finite Difference Schemes and Partial Differential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2004.
Trefethen, L. N. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Unpublished lecture notes, 1996.
———. Spectral Methods in MATLAB®. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
Trefethen, L. N., and D. Bau, III. Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997.
Trottenberg, U., C. Oosterlee, and A. Schüller. Multigrid. Burlington, MA: Academic Press, 2001.
Van der Vorst, H. A. Iterative Krylov Methods for Large Linear Systems. Cambridge, UK: Cambridge University Press, 2003.
Van Loan, C. F. Computational Frameworks for the Fast Fourier Transform. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992.
Watkins, D. Fundamentals of Matrix Computations. New York, NY: John Wiley & Sons, 2002.
Whitham, G. B. Linear and Nonlinear Waves. New York, NY: John Wiley & Sons, 1974.
