Numerical Methods

1. Solution of Partial Differential Equations (PDEs)
   1. Basic Concepts
      1. Classification of PDEs
         1. Elliptic PDEs
            1. Characteristics
            2. Example: Laplace's Equation
         2. Parabolic PDEs
            1. Characteristics
            2. Example: Heat Equation
         3. Hyperbolic PDEs
            1. Characteristics
            2. Example: Wave Equation
      2. Boundary and Initial Conditions
         1. Dirichlet Boundary Conditions
         2. Neumann Boundary Conditions
         3. Mixed Boundary Conditions
         4. Initial Conditions for Time-Dependent Problems
   2. Methods
      1. Finite Difference Method (FDM)
         1. Discretization Techniques
            1. Forward, Backward, and Central Differences
            2. Grid Point Selection
         2. Stability and Convergence
            1. Courant-Friedrichs-Lewy (CFL) Condition
            2. Von Neumann Stability Analysis
         3. Applications
            1. Solving Laplace's and Poisson's Equations
            2. Time-Stepping Schemes for Parabolic Equations
      2. Finite Element Method (FEM)
         1. Mesh Generation
            1. Structured vs. Unstructured Meshes
            2. Adaptive Mesh Refinement (AMR)
         2. Element Approximation
            1. Linear and Higher-Order Elements
            2. Shape Functions
         3. Assembly of System of Equations
            1. Global Stiffness Matrix
            2. Boundary Condition Implementation
         4. Applications
            1. Elasticity Problems
            2. Complex Geometries
      3. Finite Volume Method (FVM)
         1. Control Volume Integration
         2. Flux Calculation at Cell Interfaces
         3. Conservation Principles
         4. Applications in Fluid Dynamics
            1. Compressible and Incompressible Flows
      4. Spectral Methods
         1. Basis Functions and Expansion
            1. Fourier Series
            2. Chebyshev Polynomials
         2. Transform Techniques
            1. Fast Fourier Transform (FFT)
         3. Application in Fluid Dynamics and Quantum Mechanics
   3. Stability, Consistency, and Convergence
      1. Lax Equivalence Theorem
         1. Relation Between Consistency and Stability for Convergence
      2. Analyzing Consistency and Order of a Method
      3. Error Estimation and Control
         1. Global vs. Local Errors
         2. Adaptive Time Stepping and Mesh Refinement
   4. Handling Complex Boundaries and Domains
      1. Domain Decomposition
         1. Overlapping and Non-overlapping Techniques
      2. Immersed Boundary Methods
         1. Treatment of Irregular Boundaries
         2. Application in Biological Fluid Dynamics
      3. Multiscale and Multiphysics Problems
         1. Coupling Different Physics in a Unified Framework
         2. Example: Thermal-Structural Interaction