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Numerical Methods for Partial Differential Equations

3-0-9 units · Graduate

Covers the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential and integral equations. Topics include mathematical formulations; finite difference, finite volume, finite element, and boundary element discretization methods; and direct and iterative solution techniques. The methodologies described form the foundation for computational approaches to engineering systems involving heat transfer, solid mechanics, fluid dynamics, and electromagnetics. Computer assignments requiring programming.

Spring 2003 · Numerical Methods for Partial Differential Equations (SMA 5212)

Topics

  • Overview
  • Finite Differences: Elliptic Problems
  • Finite Differences: Parabolic Problems
  • Finite Differences: Eigenvalue, 2D Problems
  • Solution Methods: Iterative Methods
  • Solution Methods: Multigrid Methods
  • Finite Differences: Hyperbolic Problems
  • Finite Volumes: Linear Problems
  • Finite Volumes: Conservation Laws
  • Finite Volumes: Nonlinear Problems
  • Finite Elements: Variational Formulation
  • Finite Elements: Poisson 1D – I
  • Finite Elements: Poisson 1D – II
  • Finite Elements: Poisson 2D – I
  • Finite Elements: Poisson 2D – II
  • Finite Elements: General Elliptic Problems – Overview
  • Finite Elements: Parabolic Problems, Eigenvalue Problems
  • Integral Equations: Derivation
  • Integral Equations: Collocation and Galerkin Methods
  • Integral Equations: Convergence Theory – 2nd Kind
  • Integral Equations: Quadrature and Cubature
  • Integral Equations: Nystrom Methods
  • Integral Equations: Convergence Theory – 1st Kind
  • Integral Equations: Fast Solvers

Readings

  • Trefethen, L. N., and D. Bau, III. Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997. ISBN: 9780898713619.
    Listed reading
  • Numerical Methods for Conservation Laws , R. Levecque, Lectures in Mathematics, ETH Zurich, Birkhauser.
    Listed reading
  • Strang, G., and G. J. Fix. Analysis of the Finite Element Method . Upper Saddle River, NJ: Prentice-Hall, 1973. ISBN: 9780130329462.
    Listed reading
  • Quarteroni, A., and A. Valli. Numerical Approximation of Partial Differential Equations . Berlin; New York, NY: Springer-Verlag, 1997. ISBN: 9783540571117 (Berlin: acid-free paper) and ISBN: 9780387571119 (New York: acid-free paper).
    Listed reading
  • Atkinson, K. E. The Numerical Solution of Integral Equations of the Second Kind . Cambridge, UK: Cambridge University Press, 1997. ISBN: 9780521583916 (hardcopy).
    Listed reading
  • Briggs, W. L., et al. A Multigrid Tutorial . 2nd ed. Philadelphia, PA: SIAM, 2000. ISBN: 9780898714623.
    Listed reading
  • Tveito, A., and R. Winther. Introduction to Partial Differential Equations: A Computational Approach . New York, NY: Springer, 1998. ISBN: 9780387983271. (Texts in Applied Mathematics 29, are on reserve at MIT’s Barker Library and at NUS’s Central Library and SMA Library.)
    Listed reading

Prerequisites & course information

Same subject as 2.097[J] , 6.7330J Prereq: 18.03 or 18.06 G (Fall) 3-0-9 units

Sources