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Mathematical Methods for Engineers II

Lecture 12: Matrices in Difference Equations (1D, 2D, 3D)

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Course Description :

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

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