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### Outline2005

Course: MATH 511, Winter 2008
School: BYU
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511 Numerical MATH Methods for Partial Differential Equations Winter Semester 2005 Professor: Vianey Villamizar Office: 366 TMCB Class: 10:00 - 10:50 p.m. MWF 323 TMCB Email/Phone: vianey@math.byu.edu / 422-1754 Web page: www.math.byu.edu/~vianey Office Hours: M 5:00-6:00 (office), Fr 2:00-3:00 (Math lab) or by appointment. Text: Numerical Solution of PDE, Joe Flaherty's manuscript notes 1999. OUTLINE 1.-...

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511 Numerical MATH Methods for Partial Differential Equations Winter Semester 2005 Professor: Vianey Villamizar Office: 366 TMCB Class: 10:00 - 10:50 p.m. MWF 323 TMCB Email/Phone: vianey@math.byu.edu / 422-1754 Web page: www.math.byu.edu/~vianey Office Hours: M 5:00-6:00 (office), Fr 2:00-3:00 (Math lab) or by appointment. Text: Numerical Solution of PDE, Joe Flaherty's manuscript notes 1999. OUTLINE 1.- Introduction. 1.1 Example of problems leading to partial differential equations. 1.2 Classification of partial differential equations. 1.3 Hyperbolic conservation laws. Shock waves. 2.- Introduction to Finite Difference Methods. 2.1 Introduction to finite difference approximations. 2.2 Constructing difference operators. 2.3 A simple difference schemes for a kinematic wave equation. 2.3 A simple difference schemes for the heat equation. 3.- Basic Theoretical Concepts for Time-Dependent Problems. Stability, Consistency and Convergence. 3.1 Consistency, convergence and stability. 3.2 Stability analysis using a discrete Fourier series. 3.3 Matrix stability analysis. 3.4 The Lax equivalence theorem. 4. Difference Methods for Hyperbolic Partial Differential Equations. 4.1 Vector systems and characteristics. 4.2 The Courant, Friedrichs, Lewy theorem. 4.3 Dissipation and dispersion. 4.4 Multidimensional hyperbolic problems. 4.5 Acoustic scattering problems. 4.6 Finite difference time-domain method and scattering cross section. 5.- Difference Methods for Parabolic Partial Differential Equations. 5.1 Explicit and implicit finite difference schemes. 5.2 Neumann boundary conditions. 5.3 Variable coefficients. 5.4 Convection-diffusion problems. 5.5 Non-linear problems. 5.6 Multidimensional parabolic problems. ADI. 5.7 Operator splitting methods. 5.8 Polar coordinates. 6. - Elliptic Problems. 6.1 Difference schemes for Poisson's equation. 6.2 Boundary conforming coordinates. Automatic grid generation. 6.3 Solution techniques for elliptic problems. Direct solution methods. 6.4 Iterative solution methods: Jacobi, Gauss-Siedel and SOR. 7. Special Topics: Finite Element Method and/or Others. REFERENCE LIST 1.- C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Vol. 1, 2nd edition, Springer Series in Comp. Physics, 1991. 2.- J. C. Strikwerda, Finite Difference Schemes and Partial Diferential Equations, Second Edition, SIAM, 2004. 3.- A. R. Mitchell and D. E. Griffiths, The Finite Difference Method in Partial Differential Equations, J. Wiley & Sons, 1980. 4.- G. D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods , 2nd edition, Oxford Univ. Press,, 1978. 5.- J. W. Thomas, Numerical Partial Differential Finite Equations Difference Methods, Text in Appl. Math # 22, Springer, 1995. 6.- D. A. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Series in Comp. Meth. in Mech. and Thermal Scs., Hemisphere Publishing Corp., 1984. 7.- W. F. Ames, Numerical Methods for Partial Differential Equations, 3rd edition, Academic Press, 1992. 8.- G. Evans, J. Blackledge and P. Yardley, Numerical Methods for Partial Differential Equations, Springer, 2000. 9.- C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987. Course grade Course grade will be determined as follows: Midterm project Homework Final Project Projects Projects and homework will consists on some theoretical questions and applications of the numerical methods learned in class to some problems. You will need to use a computer to implement the algorithms needed for the homework or projects. I expect that you have good programming skills. Programming is an important part of this class. I strongly recommend that you use MATLAB as your programming language, but you can also use FORTRAN, JAVA, or C++. You need to choose a classmate to form a team of two students to work in the homework. However, the Midterm and Final projects need to be done individually. Regarding homework, a different team will be responsible for presenting some solutions, collecting and then grading it. Tentative Schedule. HWK will be complemented with other problems Week 1 2 3 4 5 6 7 Sections-Outline 1.1,1.2,1.3 1.3,2.1,2.2 2.3, 3.1 3.1, 3.2 3.3,...

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