Pre- Requisite: Numerical Methods

Introduction:

This course addresses post graduate students of all fields who are interested in numerical methods for partial differential equations, with focus on a rigorous mathematical basis. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. Of particular focus are a qualitative understanding of the considered partial differential equation, fundamentals of finite difference, finite volume, finite element, and spectral methods, and important concepts such as stability, convergence, and error analysis. Upon completion, students are able to solve following problems: (1) advection equation, heat equation, wave equation, Airy equation, convection-diffusion problems, KdV equation, hyperbolic conservation laws, Poisson equation, Stokes problem, Navier-Stokes equations, interface problems. (2) consistency, stability, convergence, Lax equivalence theorem, error analysis, Fourier approaches, staggered grids, shocks, front propagation, preconditioning, multi-grid, Krylov spaces, saddle point problems. (3) Finite differences, finite volumes, finite elements, ENO/WENO, spectral methods, projection approaches for incompressible owes, level set methods, particle methods, direct and iterative methods, multi-grid.

Learning Outcomes:

On successful completion of this course students will be able to:

Use knowledge of partial differential equations (PDEs),

Modelling, the general structure of solutions, and analytic and numerical methods for solutions.

Formulate physical problems as PDEs using conservation laws.

 Course Contents:

1. Boundary and initial conditions,
2. Polynomial approximations in higher dimensions,
3. Finite Element Method: The Galerkin method in one and more dimensions,
4. Error bound on the Galarki method, the method of collocation,
5. error bounds on the collocation method,
6. Comparison of efficiency of the finite difference and finite element method,
7. Finite Difference Method: Finite difference approximations,
8. Application to solution of linear and non-linear partial differential equations appearing in physical problem

Recommended Books

1. Silvia, B., Silvia, F., Giovanni R., Chi-Wang, S.,Numerical Solutions of Partial Differential Equations (Birkhäuser Basel (2008)).
2. Claes Johnson. Numerical Solutions Of Partial Differential Equations By The Finite Element Method (Dover Publications (2009)).

Suggested Books

1. Morton, K.W. and Mayers, D.F.,Numerical solution of partial differential equations (Cambridge University Press (2005)).
2. Roe, P. L. andChattot. J. J., Innovative methods for numerical solutions of partial differential equations (World Scientific (2002)).
3. Alan Jeffrey. Applied partial differential equations. An introduction (Academic Press (2002))
4. Luis, A. C., François, G., Yan, G., Carlos, E.K., Alexis, V.,Nonlinear Partial Differential Equations (Birkhäuser Basel (2012)).
5. Rafael, José Iorio Jr_ Valéria de MagalhãesIorio. Fourier analysis and partial differential equations (Cambridge University Press (2001)).
6. Recent Published Research Papers

Assignment Criteria: 

Mid Term Marks: 30

Final Term Marks:50

Sessional Marks: 20

Key Dates and Time of Class Meeting

Tuesday: 2:00 p.m to 3:30 p.m

Wednseday: 3:30 to 5: 00 p.m

Commencement of Classes                                                    November 02, 2020

Mid Term Examination                                                             December 28, 2020

Final Term Examination                                                           March 01-05, 2021

Declaration of Result                                                               March 12, 2021