MEC 4677 - Applied Numerical Methods

• Identify sources of errors in numerical computation
• Develop or select a mathematical model appropriate for desired application
• Formulate proper constraints and conditions for solving the model equations
• Select and implement appropriate numerical method for solving the model equations
• Analyze, critique, discuss, and document the results from numerical simulations
• Demonstrate ability to make design decisions using results from numerical simulations

• Static equilibrium of solids, liquids, and gases
• Newton’s Laws of Motion for particle and rigid bodies
• Combined loading of shafts and structural members, beam deflection and column buckling
• First and second order electrical, mechanical, and electromechanical systems modeled by lumped parameter approach
• Response of first and second order linear time invariant systems in time domain and frequency domain
• First and Second law of thermodynamics
• Euler, Bernoulli, and Navier-Stokes equations for fluid flow
• Laws of heat transfer
• Hydrodynamic and thermal boundary layers

• Sources of errors in numerical computation
• Fixed point iteration, root finding methods and convergence rates for a single algebraic equation
• Numerical linear algebra: solving system of linear algebraic equations, and matrix eigenvalues
• Solving system of nonlinear algebraic equations
• Lagrange/Newton interpolation, and cubic splines
• Numerical differentiation, optimal step size, Richardson extrapolation
• Numerical integration and quadrature methods
• Numerical solution for ordinary differential equations (ODEs): initial value problems (IVP) and boundary value problems (BVP), stability of numerical solution for stiff ODEs, eigenvalues
• Numerical solution to partial differential equations (PDEs) using finite eifference and finite volume methods, introduction to finite element method (if time permits)
• Engineering applications of numerical methods

• Fixed point iteration and root finding
• System of linear algebraic equations
• Eigenvalues and eigenvectors
• System of nonlinear algebraic equations
• Interpolation and splines
• Numerical differentiation and Gauss quadrature
• Step-size influence on stability for methods of solving single first order ODE (IVP)
• Numerical methods of solving system of first order ODEs (IVP)
• Shooting method for solving ODEs (BVP)
• Finite Difference Method for solving ODEs (BVP)
• Numerical solution for parabolic PDEs (IBVP)
• Numerical solution for elliptic PDEs (BVP)
• Numerical solution for hyperbolic PDEs (IBVP)