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Numerical Methods in Civil Engineering

Lecture 1: Mod-01 Lec-01 Lecture-01-Introduction to Numerical Methods

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Lecture Details :

Numerical Methods in Civil Engineering by Prof.Arghya Deb, Department of Civil Engineering,IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in

Course Description :

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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