MTH-206.4 Computational PDE

• Review of basic numerical analysis

• Finite differences for linear equations

Linear hyperbolic equations, finite differences, theoretical concepts of stability and consistency, order of accuracy, upwind, Lax-Fredrichs and Lax-Wendroff schemes.

Linear parabolic equations-explicit and implicit schemes, Crank-Nicholson method, introduction to multi-dimensional problems.

Linear elliptic equations-finite difference schemes.

• Finite Difference schemes for nonlinear equations

One dimensional scalar conservation laws, review of basic theory, solutions of the Riemann problem and entropy conditions. First order schemes like Lax Fredrichs, Godunov, Enquist Osher and Roe's scheme. Convergence results, entropy consistency and numerical viscosity. Introduction to higher order schemes-Lax Wendroff scheme, Upwind schemes of Van Leer, ENO schemes, Central schemes, Relaxation methods. Introduction to finite volume methods. Convection-Reaction-Diffusion equations,Extension to the above methods. Splitting schemes for multi-dimensional problems.

• Finite element methods for linear equations

Review of elliptic equations, weak formulation and Lax-Milgram lemma, Galerkin approximation, basis functions, energy methods and error estimates, Cea's estimate and Babuska Brezzi theorem.

Finite elements for parabolic equations - Galerkin approximation and error estimates. A posteriori error estimates for Elliptic and Parabolic equations.

• Spectral Methods

Fast Fourier transformation, introduction to Fourier, spectral and pseudo spectral methods.

• Implementation of algorithms on computers is an integrable part of this course.