Numerical Methods and Modelling Week 7-12 Notes

Description

Week 7: Numerical Integration – trapezoidal rule (first-order), Simpson’s rule (second-order), step-size control, adaptive accuracy, and polynomial approximation methods for estimating areas under curves. Week 8: Gaussian Quadrature – integration using weighted sampling points (roots of Legendre polynomials), higher accuracy with fewer evaluations, no need for evenly spaced data; used in FEM and engineering analysis. Week 9: Roots of Non-Linear Equations – graphical, stepping, bisection, Newton–Raphson, and secant methods; error tolerance, convergence behaviour, and MATLAB fzero application. Week 10: Optimization – downhill (search) method, golden-section search, quadratic approximation, and practical example (rocket-launch range). Emphasises finding local vs. global minima and algorithm efficiency. Week 11: Ordinary Differential Equations (Initial-Value Problems) – Euler’s and Runge–Kutta methods, truncation error and adaptive step sizing, error monitoring through step-doubling. Week 12: Boundary-Value Problems & Finite Difference Method – solving ODEs with conditions at multiple points; discretisation into nodes, matrix formulation, examples (pressure vessel, beam deflection, heat transfer); linear vs. non-linear systems, geometric and material non-linearities, iterative convergence.