Introduction to Numerical Methods:
- Why study numerical methods.
- Sources of error in numerical solutions: truncation error, round off error.
- Order of accuracy - Taylor series expansion.
Direct Solution of Linear systems:
- Gauss elimination, Gauss Jordan elimination.
- Pivoting, inaccuracies due to pivoting.
- Factorization, Cholesky decomposition.
- Diagonal dominance, condition number, ill conditioned matrices, singularity and singular value decomposition.
- Banded matrices, storage schemes for banded matrices, skyline solver.
Iterative solution of Linear systems:
- Jacobi iteration.
- Gauss Seidel iteration.
- Convergence criteria.
Direct Solution of Non Linear systems:
- Newton Raphson iterations to find roots of a 1D nonlinear equation.
- Generalization to multiple dimensions.
- Newton Iterations, Quasi Newton iterations.
- Local and global minimum, rates of convergence, convergence criteria.
Iterative Solution of Non Linear systems:
- Conjugate gradient.
- Preconditioning.
Partial Differential Equations:
- Introduction to partial differential equations.
- Definitions & classifications of first and second order equations.
- Examples of analytical solutions.
- Method of characteristics.
Numerical Differentiation:
- Difference operators (forward, backward and central difference).
- Stability and accuracy of solutions.
- Application of finite difference operators to solve initial and boundary value problems.
Introduction to the Finite Element Method as a method to solve partial differential equations:
- Strong form of the differential equation.
- Weak form.
- Galerkin method: the finite element approximation.
- Interpolation functions: smoothness, continuity, completeness, Lagrange polynomials.
- Numerical quadrature: Trapezoidal rule, simpsons rule,Gauss quadrature.
Numerical integration of time dependent partial differential equations:
- Parabolic equations: algorithms - stability, consistency and convergence, Lax equivalence theorem.
- Hyperbolic equations: algorithms - Newmark's method,stability and accuracy, convergence, multi-step methods.
Numerical solutions of integral equations:
- Types of integral equations.
- Fredholm integral equations of the first and second kind.
- Fredholm's Alternative theorem.
- Collocation and Galerkin methods for solving integral equations.
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