Introduction to Numerical Simulation (SMA 5211)

Topics

• Example Problems and Basic Equations
• Equation Formulation Methods - Stamping Techniques, Nodal versus Node-Branch Form
• Linear System Solution - Dense GE, Conditioning, Stability
• Direct Methods for Sparse Linear Systems - Data Structures, Fill-in, Ordering, Graph Interpretations
• Linear System Solution - Orthogonalization Methods, QR, Singular Matrices
• QR and Krylov Iterative Methods. Brief Convergence Analysis.
• Krylov Methods (cont.)
• Nonlinear System Solution - 1D Newton Methods, Convergence Analysis
• Nonlinear System Solution-Multi-D Newton, Forming Jacobian by Stamping Approach, Singularity
• Nonlinear System Solution - Damping, Optimization and Continuation Schemes
• Nonlinear System Solution - Matrix-Implicit Methods and Methods for Singular Problems
• ODE Solution Methods - BE, FE, Trap Examples, Convergence
• ODE Solution Methods - Multistep Methods and Stability, Runga-Kutta Methods
• ODE Solution Methods - Stiffly Stable and Conservative Schemes
• Time-Periodic Solution Methods - Finite-Difference and Shooting Methods
• Time-Periodic Solution Methods - Matrix-Implicit Algorithms and Preconditioning
• Molecular Dynamics - Basic Numerical Issues
• Molecular Dynamics (cont.)
• 3-D Elliptic Problems - F-D Methods, Error Estimation
• 3-D Elliptic Problems - Finite-Element and Spectral Methods
• 3-D Elliptic Problems - FFT and Multigrid Methods
• 3-D Elliptic Problems - Boundary-Element Approach
• 3-D Elliptic Problems - FFT and Multipole Methods
• Model Order Reduction I
• Model Order Reduction II
• Class Choice