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Readings

Readings are assigned in the following textbooks:

[Evans] = Amazon logo Evans, Lawrence C. Partial Differential Equations. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.

[Strang] = Amazon logo Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.

[LeVeque2007] = Amazon logo LeVeque, Randall J. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007. ISBN: 9780898716290.

[Trefethen] = Amazon logo Trefethen, Lloyd N. Spectral Methods in MATLAB (Software, Environments, Tools). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001. ISBN: 9780898714654.

[LeVeque2002] = Amazon logo LeVeque, Randall J. Finite Volume Methods for Hyperbolic Problems. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521009249.

[Fletcher] = Amazon logo 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] = Amazon logo 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 # TOPICS READINGS
1 Fundamental concepts and examples Evans 1.1, 1.2
2 Well-posedness and Fourier methods for linear initial value problems

Evans 1.3

Strang 6.1
3 Laplace and Poisson equation Evans 2.2
4 Heat equation, transport equation, wave equation Evans 2.1 and 2.3-2.4
5 General finite difference approach and Poisson equation LeVeque2007, chapter 1, 2.1-2.4, 2.12
6 Elliptic equations and errors, stability, Lax equivalence theorem LeVeque2007 2.5-2.9, 2.15 (recommended 2.17)
7 Spectral methods Trefethen, chapters 1, 3
8 Fast Fourier transform (guest lecture by Stephen Johnson)
9 Spectral methods Trefethen, chapters 6, 7, 8
10 Elliptic equations and linear systems LeVeque2007, chapter 3
11 Efficient methods for sparse linear systems: multigrid

LeVeque2007 4.1-4.2, 4.6

Strang 7.1
12 Efficient methods for sparse linear systems: Krylov methods LeVeque2007 4.3-4.4
13 Ordinary differential equations LeVeque2007 9.2, chapter 5
14 Stability for ODE and von Neumann stability analysis LeVeque2007, chapter 7, 9.6-9.7
15 Advection equation and modified equation LeVeque2002 4.4-4.9, 8.6
16 Advection equation and ENO/WENO LeVeque2002 6.1, 6.3, 6.7, 10.4
17 Conservation laws: theory LeVeque2002, chapter 11
18 Conservation laws: numerical methods LeVeque2002 12.1-12.9, (recommended 12.10-12.11)
19 Conservation laws: high resolution methods LeVeque2002 12.12, 6.4-6.12
20 Operator splitting, fractional steps
21 Systems of IVP, wave equation, leapfrog, staggered grids

Notes on the wave equation (This resource may not render correctly in a screen reader.PDF)

Notes on perfectly matching layers (This resource may not render correctly in a screen reader.PDF)

(Courtesy of Steven G. Johnson. Used with permission.)
22 Level set method

Notes on the level set method (PDF)

Presentation on the level set method (PDF)

(Courtesy of Per Olof Persson. Used with permission.)
23 Navier-Stokes equation: finite difference methods

Fletcher, Canuto
(no spefic section) 
24 Navier-Stokes equation: pseudospectral methods Fletcher, Canuto
(no spefic section)
25 Particle methods
26 Project presentations