18.336
Fast Methods for Partial Differential and Integral Equations
Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.
Spring 2009 · Numerical Methods for Partial Differential Equations
Topics
- Fundamental concepts and examples
- Well-posedness and Fourier methods for linear initial value problems
- Laplace and Poisson equation
- Heat equation, transport equation, wave equation
- General finite difference approach and Poisson equation
- Elliptic equations and errors, stability, Lax equivalence theorem
- Spectral methods
- Fast Fourier transform (guest lecture by Stephen Johnson)
- Elliptic equations and linear systems
- Efficient methods for sparse linear systems: multigrid
- Efficient methods for sparse linear systems: Krylov methods
- Ordinary differential equations
- Stability for ODE and von Neumann stability analysis
- Advection equation and modified equation
- Advection equation and ENO/WENO
- Conservation laws: theory
- Conservation laws: numerical methods
- Conservation laws: high resolution methods
- Operator splitting, fractional steps
- Systems of IVP, wave equation, leapfrog, staggered grids
- Level set method
- Navier-Stokes equation: finite difference methods
- Navier-Stokes equation: pseudospectral methods
- Particle methods
- Project presentations
Readings
- [Evans] = Evans, Lawrence C. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.
- [Fletcher] = Fletcher, C. A. J. Computational Techniques for Fluid Dynamics. Fundamental and General Techniques Volume I . Springer series in computational physics. New York, NY: Springer-Verlag, 1996. ISBN: 9783540530589.
- [Canuto] = Canuto, Claudio S., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics . New York, NY: Springer-Verlag, 2007. ISBN: 9783540307273.
- SES #: 1 · TOPICS: Fundamental concepts and examples · READINGS: Evans 1.1, 1.2
- SES #: 3 · TOPICS: Laplace and Poisson equation · READINGS: Evans 2.2
- SES #: 4 · TOPICS: Heat equation, transport equation, wave equation · READINGS: Evans 2.1 and 2.3-2.4
- SES #: 5 · TOPICS: General finite difference approach and Poisson equation · READINGS: LeVeque2007, chapter 1, 2.1-2.4, 2.12
- SES #: 6 · TOPICS: Elliptic equations and errors, stability, Lax equivalence theorem · READINGS: LeVeque2007 2.5-2.9, 2.15 (recommended 2.17)
- SES #: 7 · TOPICS: Spectral methods · READINGS: Trefethen, chapters 1, 3
- SES #: 9 · TOPICS: Spectral methods · READINGS: Trefethen, chapters 6, 7, 8
- SES #: 10 · TOPICS: Elliptic equations and linear systems · READINGS: LeVeque2007, chapter 3
- SES #: 12 · TOPICS: Efficient methods for sparse linear systems: Krylov methods · READINGS: LeVeque2007 4.3-4.4
- SES #: 13 · TOPICS: Ordinary differential equations · READINGS: LeVeque2007 9.2, chapter 5
- SES #: 14 · TOPICS: Stability for ODE and von Neumann stability analysis · READINGS: LeVeque2007, chapter 7, 9.6-9.7
- SES #: 15 · TOPICS: Advection equation and modified equation · READINGS: LeVeque2002 4.4-4.9, 8.6
- SES #: 16 · TOPICS: Advection equation and ENO/WENO · READINGS: LeVeque2002 6.1, 6.3, 6.7, 10.4
- SES #: 17 · TOPICS: Conservation laws: theory · READINGS: LeVeque2002, chapter 11
- SES #: 18 · TOPICS: Conservation laws: numerical methods · READINGS: LeVeque2002 12.1-12.9, (recommended 12.10-12.11)
- SES #: 19 · TOPICS: Conservation laws: high resolution methods · READINGS: LeVeque2002 12.12, 6.4-6.12
- SES #: 23 · TOPICS: Navier-Stokes equation: finite difference methods · READINGS: Fletcher, Canuto (no spefic section)
- SES #: 24 · TOPICS: Navier-Stokes equation: pseudospectral methods · READINGS: Fletcher, Canuto (no spefic section)
- Fletcher, C. A. J. Computational Techniques for Fluid Dynamics. Fundamental and General Techniques Volume I . Springer series in computational physics. New York, NY: Springer-Verlag, 1996. ISBN: 9783540530589.
- ———. Computational Techniques for Fluid Dynamics. Specific Techniques for Different Flow Categories Volume II . Springer series in computational physics. New York, NY: Springer-Verlag, 1991. ISBN: 9783540536017.
- Canuto, Claudio S., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics . New York, NY: Springer-Verlag, 2007. ISBN: 9783540307273.
- Evans, Lawrence C. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.
Prerequisites & course information
Same subject as 6.7340J Prereq: 6.7300[J] , 16.920[J] , 18.085 , 18.335[J] , or permission of instructor G (Spring) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2009
- MIT OCW calendar ↗
- MIT OCW readings ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.