everything.study
18.086

Computational Science and Engineering II

3-0-9 units · Undergraduate

Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.

Spring 2006 · Mathematical Methods for Engineers II

Topics

  • Differential Equations and Stable Difference Methods
  • Finite Differences: Accuracy, Stability, Convergence
  • The Wave Equation and von Neumann Stability
  • The Heat Equation and Convection-Diffusion
  • Maxwell’s Equations and Staggered Leapfrog
  • Nonlinear Flow Equations
  • Separation of Variables and Spectral Methods
  • Elimination with Reordering
  • Iterative Methods
  • Multigrid Methods
  • Krylov Methods
  • Saddle Points and the Stokes Problem
  • Optimization

Readings

  • Strang, Gilbert. Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press , 1986. ISBN: 9780961408800. ( Table of Contents ) ↗
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  • ———. Computational Science and Engineering . Wellesley, MA: Wellesley-Cambridge Press , 2007. ISBN: 9780961408817. ↗
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  • Draft versions of chapters relevant to this course are available below.
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  • Anderson, E., et al. LAPACK User’s Guide . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999.
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  • 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.
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  • Axelsson, O. Iterative Solution Methods . Cambridge, UK: Cambridge University Press, 1994.
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  • Axelsson, O., and V. A. Barker. Finite Element Solution of Boundary Value Problems . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001.
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  • Bathe, K. J. Finite Element Procedures . Upper Saddle River, NJ: Prentice Hall, 1996.
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  • Bender, C. M., and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers . New York, NY: McGraw-Hill, 1978.
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  • Björck, A. Numerical Methods for Least Squares Problems . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996.
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  • Bracewell, R. N. The Fourier Transform and Its Applications . New York, NY: McGraw-Hill, 1986.
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  • Brenner, S. C., and L. R. Scott. The Mathematical Theory of Finite Element Methods . New York, NY: Springer-Verlag, 1994.
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  • Brezzi, F., and M. Fortin. Mixed and Hybrid Finite Element Methods . New York, NY: Springer-Verlag, 1991.
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  • Briggs, W. L., V. Henson, and S. F. McCormick. A Multigrid Tutorial . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
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  • Butcher, J. C. The Numerical Analysis of Ordinary Differential Equations . New York, NY: John Wiley & Sons, 1987.
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  • Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics . New York, NY: Springer-Verlag, 1987.
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  • Ciarlet, P. G. The Finite Element Method for Elliptic Problems . Amsterdam, NL: North-Holland Pub. Co., 1978.
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  • 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.
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  • Elman, H., D. Silvester, and A. Wathen. Finite Elements and Fast Iterative Solvers . New York, NY: Oxford University Press, 2005.
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  • Fornberg, B. A Practical Guide to Pseudospectral Methods . Cambridge, UK: Cambridge University Press, 1996.
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  • Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations . Upper Saddle River, NJ: Prentice Hall, 1971.
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  • George, A., and J. W. Liu. Computer Solution of Large Sparse Positive Definite Systems . Upper Saddle River, NJ: Prentice Hall, 1981.
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  • 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.
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  • Golub, G. H., and C. F. Van Loan. Matrix Computations . Baltimore, MD: Johns Hopkins University Press, 1996.
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  • Greenbaum, A. Iterative Methods for Solving Linear Systems . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997.
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  • Gresho, P., and R. Sani. Incompressible Flow and the Finite Element Method . Volume 1: Advection-Diffusion . New York, NY: John Wiley & Sons, 1998.
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  • Griffiths, D. F., and G. A. Watson, eds. Numeral Analysis . New York, NY: Longmans, 1986.
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  • Gustafsson, B., H. -O. Kreiss, and J. Oliger. Time-Dependent Problems and Difference Methods . New York, NY: John Wiley & Sons, 1995.
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  • Hackbusch, W. Multigrid Methods and Applications . New York, NY: Springer-Verlag, 1985.
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  • Hackbusch, W., and U. Trottenberg. Multigrid Methods . Lecture Notes in Math . Vol. 960. New York, NY: Springer-Verlag, 1982.
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  • Hairer, E., S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems . New York, NY: Springer-Verlag, 1987.
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  • Hairer, E., and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems . New York, NY: Springer-Verlag, 1991.
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  • Henrici, P. Discrete Variable Methods in Ordinary Differential Equations . New York, NY: John Wiley & Sons, 1962.
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  • ———. Applied and Computational Complex Analysis, III . New York, NY: John Wiley & Sons, 1986.
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  • Higham, D. J., and N. J. Higham. MATLAB® Guide . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2005.
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  • Higham, N. J. Accuracy and Stability of Numeral Algorithms . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996.
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  • Hughes, T. J. R. The Finite Element Method . Upper Saddle River, NJ: Prentice Hall, 1987.
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  • Iserles, A. A First Course in the Numerical Analysis of Differential Equations . Cambridge, UK: Cambridge University Press, 1996.
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  • Johnson, C. Numerical Solutions of Partial Differential Equations by The Finite Element Method . Cambridge, UK: Cambridge University Press, 1987.
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  • Keller, H. B. Numerical Solution of Two Point Boundary Value Problems . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1976.
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  • Kelley, C. T. Iterative Methods for Optimization . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999.
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  • Lambert, J. D. Numerical Methods for Ordinary Differential Systems . New York, NY: John Wiley & Sons, 1991.
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  • Lax, P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1973.
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  • LeVeque, R. J. Numerical Methods for Conservation Laws . 2nd ed. Boston, MA: Birkhäuser Verlag, 1992.
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  • Mallat, S. A Wavelet Tour of Signal Processing . Burlington, MA: Academic Press, 1999.
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  • Oppenheim, A. V., and R. W. Schafer. Discrete-Time Signal Processing . Upper Saddle River, NJ: Prentice Hall, 1989.
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  • Parlett, B. N. The Symmetric Eigenvalue Problem . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1998.
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  • Petzold, L. R. “A Description of DASSL - A Differential Algebraic System Solver.” IMACS Trans Sci Comp 1 (1982): 1-65. North Holland.
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  • Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes . Cambridge, UK: Cambridge University Press, 1986.
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  • Quarteroni, A., and A. Valli. Numerical Approximation of Partial Differential Equations . New York, NY: Springer-Verlag, 1997.
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  • Richtmyer, R. D., and K. W. Morton. Difference Methods for Initial-Value Problems . New York, NY: John Wiley & Sons, 1967.
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  • Saad, Y. Iterative Methods for Sparse Linear Systems . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2003.
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  • Shampine, L. F., and C. W. Gear. “A User’s View of Solving Stiff Ordinary Differential Equations.” SIAM Review 21 (1979): 1-17.
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  • Sod, G. A. Numerical Methods in Fluid Dynamics . Cambridge, UK: Cambridge University Press, 1985.
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  • Stoer, J., and R. Bulirsch. Introduction to Numerical Analysis . New York, NY: Springer-Verlag, 2002.
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  • Strang, G. Introduction to Linear Algebra . Wellesley, MA: Wellesley-Cambridge Press, 2003.
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  • Strang, G., and K. Borre. Linear Algebra, Geodesy, and GPS . Wellesley, MA: Wellesley-Cambridge Press, 1997.
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  • 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.
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  • Strang, G., and T. Nguyen. Wavelets and Filter Banks . Wellesley, MA: Wellesley-Cambridge Press, 1996.
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  • Strikwerda, J. C. Finite Difference Schemes and Partial Differential Equations . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2004.
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  • Trefethen, L. N. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations . Unpublished lecture notes, 1996.
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  • ———. Spectral Methods in MATLAB® . Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
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  • Trefethen, L. N., and D. Bau, III. Numerical Linear Algebra . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997.
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  • Trottenberg, U., C. Oosterlee, and A. Schüller. Multigrid . Burlington, MA: Academic Press, 2001.
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  • Van der Vorst, H. A. Iterative Krylov Methods for Large Linear Systems . Cambridge, UK: Cambridge University Press, 2003.
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  • Van Loan, C. F. Computational Frameworks for the Fast Fourier Transform . Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992.
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  • Watkins, D. Fundamentals of Matrix Computations . New York, NY: John Wiley & Sons, 2002.
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  • Whitham, G. B. Linear and Nonlinear Waves . New York, NY: John Wiley & Sons, 1974.
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  • This course as taught during the Spring 2006 term on the MIT campus used the following text:
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  • Since that time, Professor Strang has published a new textbook that is being used for this course as it is currently taught on the MIT campus, as well as for Mathematical Methods for Engineers I (18.085). Information about the new book can be found at the Wellesley-Cambridge Press Web site, along with a link to Prof. Strang’s new “Computational Science and Engineering” Web page developed as a resource for everyone learning and doing Computational Science and Engineering.
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  • Strang, Gilbert. Computational Science and Engineering . Wellesley, MA: Wellesley-Cambridge Press , 2007. ISBN: 9780961408817. ↗
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Prerequisites & course information

Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Spring) Not offered regularly; consult department 3-0-9 units

Sources