UsefulLinks
Mathematics
Numerical Methods
1. Introduction to Numerical Methods
2. Foundational Concepts in Numerical Computation
3. Root Finding Methods
4. Linear Systems of Equations
5. Interpolation and Approximation
6. Curve Fitting and Least Squares
7. Numerical Differentiation
8. Numerical Integration
9. Ordinary Differential Equations
10. Partial Differential Equations
11. Optimization Methods
12. Eigenvalue Problems
10.
Partial Differential Equations
10.1.
PDE Classification
10.1.1.
Second-Order PDEs
10.1.2.
Elliptic Equations
10.1.2.1.
Laplace Equation
10.1.2.2.
Poisson Equation
10.1.2.3.
Boundary Conditions
10.1.3.
Parabolic Equations
10.1.3.1.
Heat Equation
10.1.3.2.
Diffusion Equations
10.1.3.3.
Initial and Boundary Conditions
10.1.4.
Hyperbolic Equations
10.1.4.1.
Wave Equation
10.1.4.2.
Transport Equations
10.1.4.3.
Characteristic Methods
10.2.
Finite Difference Methods
10.2.1.
Grid Generation
10.2.1.1.
Uniform Grids
10.2.1.2.
Nonuniform Grids
10.2.1.3.
Boundary Conforming
10.2.2.
Difference Approximations
10.2.2.1.
Central Differences
10.2.2.2.
Forward and Backward
10.2.2.3.
Higher-Order Schemes
10.2.3.
Elliptic Problems
10.2.3.1.
Five-Point Stencil
10.2.3.2.
Iterative Solution
10.2.3.3.
Multigrid Methods
10.2.4.
Parabolic Problems
10.2.4.1.
Explicit Schemes
10.2.4.2.
Implicit Schemes
10.2.4.3.
Crank-Nicolson Method
10.2.4.4.
Stability Analysis
10.2.5.
Hyperbolic Problems
10.2.5.1.
Upwind Schemes
10.2.5.2.
Lax-Wendroff Method
10.2.5.3.
CFL Condition
10.3.
Finite Element Method Introduction
10.3.1.
Weak Formulation
10.3.1.1.
Variational Principles
10.3.1.2.
Integration by Parts
10.3.1.3.
Boundary Conditions
10.3.2.
Element Types
10.3.2.1.
Linear Elements
10.3.2.2.
Quadratic Elements
10.3.2.3.
Shape Functions
10.3.3.
Assembly Process
10.3.3.1.
Element Matrices
10.3.3.2.
Global System
10.3.3.3.
Boundary Conditions
10.3.4.
Solution Techniques
10.3.4.1.
Direct Solvers
10.3.4.2.
Iterative Solvers
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9. Ordinary Differential Equations
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11. Optimization Methods