The exhaustive list of topics in Numerical Methods in Civil Engineering in which we provide Help with Homework Assignment and Help with Project is as follows:
- Numerical Methods:
- 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:
- Partial differential equations.
- First and second order equations.
- 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.
- 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.