Mathematical Methods for Engineers II

Difference Methods for Ordinary Differential Equations – Finite Differences, Accuracy, Stability, Convergence – The One-way Wave Equation and CFL / von Neumann Stability – Comparison of Methods for the Wave Equation – Second-order Wave Equation (including leapfrog) – Wave Profiles, Heat Equation / point source – Finite Differences for the Heat Equation – Convection-Diffusion / Conservation Laws – Conservation Laws / Analysis / Shocks – Shocks and Fans from Point Source – Level Set Method – Matrices in Difference Equations (1D, 2D, 3D)

Elimination with Reordering: Sparse Matrices – Financial Mathematics / Black-Scholes Equation – Iterative Methods and Preconditioners – General Methods for Sparse Systems – Multigrid Methods – Krylov Methods / Multigrid Continued – Conjugate Gradient Method – Fast Poisson Solver – Optimization with constraints – Weighted Least Squares – Calculus of Variations / Weak Form – Error Estimates / Projections – Saddle Points / Inf-sup condition – Two Squares / Equality Constraint Bu = d – Regularization by Penalty Term – Linear Programming and Duality – Duality Puzzle / Inverse Problem / Integral Equations