Numerical Methods in Civil

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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