MA 211 Numerical Methods for Partial Differential Equations

 Problem Sheets:

1. Problems related to hyperbolic PDEs

Policy of Attendance

Attendance in all lecture and tutorial classes is compulsory. 

Students, who do not meet 75% attendance requirement will not be allowed to write the end semester examination.

For attendance in the classes, attendance sheets will be circulated. 

Each student is expected to sign against his/her name only.

In case, any student is found marking proxy for some other student, an appropriate disciplinary action will be taken on both students involved in the proxy matter.

Random attendance will also be taken.

Syllabus

Review of direct and iterative methods for solving linear system of equations. Review of general features and classification of PDEs and boundary conditions.

Basic concepts of discretization. Difference equations. Errors. Explicit and implicit approaches. Consistency, stability and convergence. Analysis of stability: discrete perturbation, matrix and the von Neumann approach. Lax equivalence theorem. Numerical and analytical domains of dependence.

Numerical solutions of elliptic PDEs: Iterative methods of solutions. Finite difference solution of Poisson equation. Alternate Direction Implicit methods.

Propagation problems. The modified PDE. Implicit numerical diffusion and dispersion. Asymptotic steady-state solution of propagation problems.

Numerical solutions of parabolic PDEs: Forward Time Centered Space (FTCS), Dufort-Frankel, Backward Time Centered Space (BTCS) and Crank-Nicolson methods.

Numerical solutions of hyperbolic PDEs: Method of characteristics. FTCS, upwind, Lax, Leapfrog, BTCS, Lax-Wendroff single & multistep and MacCormack methods. Numerical methods of characteristics.

Introduction to coordinate transformation and grid generation. Introduction to multigrid methods

Texts: 

1. J.  D. Hoffman, Numerical methods for Engineers and Scientists, McGraw Hill, 1993.

2. G. D. Smith, Numerical solutions to Partial Differential Equations, Brunel University, Clarendon Press, 1985. 

3. A. R. Mitchel and D. F. Griffiths, Finite Difference Method in Partial Differential Equations, John Wiley, 1980. 

4. L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley, 1982. 

References:

1. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM  2004.

2. H. P. Langtangen, Computational Partial Differential Equations, Springer Verlag, 1999.

3. H. Roos, C. Grossmann, and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, 2007. 

4.  A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Verlag, 1994.

5. A. Tveito and R. Winther, Introduction to Partial Differential Equations: A Computational Approach, Springer, 1998.

6. R.  L. Burden and J. D. Fairs, Numerical Analysis, Brooks/Cole, 2001.

  Lecture Timings