# Numerical Methods in Civil Engineering

## IIT Kharagpur , Prof.Arghya Deb

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### Lecture Description

Numerical Methods in Civil Engineering by Dr. A. Deb,Department of Civil Engineering,IIT Kharagpur.For more details on NPTEL visit http://nptel.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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