Computational Geosciences

  1. Numerical Methods in Geosciences
    1. Discretization of Continuous Systems
      1. Grids and Meshes
        1. Structured Grids
          1. Cartesian Grids
            1. Regular Spacing
              1. Coordinate Transformations
                1. Boundary Fitting
                2. Regular Lattices
                  1. Hexagonal Grids
                    1. Triangular Grids
                      1. Grid Refinement
                    2. Unstructured Meshes
                      1. Triangular and Tetrahedral Meshes
                        1. Delaunay Triangulation
                          1. Quality Metrics
                            1. Mesh Optimization
                            2. Mesh Generation Techniques
                              1. Advancing Front Method
                                1. Quadtree/Octree Methods
                                  1. Mesh Smoothing
                                2. Adaptive Mesh Refinement
                                  1. Principles of AMR
                                    1. Error Estimation
                                      1. Refinement Criteria
                                        1. Data Structures
                                        2. Applications in Geosciences
                                          1. Shock Capturing
                                            1. Multiscale Phenomena
                                              1. Computational Efficiency
                                          2. Finite Difference Method
                                            1. Derivation of Finite Difference Approximations
                                              1. Taylor Series Expansion
                                                1. Forward Differences
                                                  1. Backward Differences
                                                    1. Central Differences
                                                    2. Stability and Consistency
                                                      1. Von Neumann Stability Analysis
                                                        1. CFL Condition
                                                          1. Convergence Analysis
                                                          2. Application to ODEs and PDEs
                                                            1. Boundary Value Problems
                                                              1. Initial Value Problems
                                                                1. Mixed Problems
                                                              2. Finite Element Method
                                                                1. Weak Formulation of Equations
                                                                  1. Variational Principles
                                                                    1. Galerkin Method
                                                                      1. Weighted Residual Methods
                                                                      2. Basis Functions and Elements
                                                                        1. Linear Elements
                                                                          1. Quadratic Elements
                                                                            1. Shape Functions
                                                                            2. Assembly and Solution of FEM Systems
                                                                              1. Element Matrices
                                                                                1. Global Assembly
                                                                                  1. Boundary Condition Implementation
                                                                                2. Finite Volume Method
                                                                                  1. Conservation Laws and Control Volumes
                                                                                    1. Integral Form of Conservation Laws
                                                                                      1. Control Volume Definition
                                                                                        1. Flux Balance
                                                                                        2. Flux Calculation
                                                                                          1. Upwind Schemes
                                                                                            1. Central Schemes
                                                                                              1. High-Resolution Schemes
                                                                                              2. Applications in Fluid Dynamics
                                                                                                1. Compressible Flow
                                                                                                  1. Incompressible Flow
                                                                                                    1. Multiphase Flow
                                                                                                  2. Spectral Methods
                                                                                                    1. Basis Functions
                                                                                                      1. Fourier Basis
                                                                                                        1. Chebyshev Polynomials
                                                                                                          1. Legendre Polynomials
                                                                                                          2. Spectral Accuracy
                                                                                                            1. Exponential Convergence
                                                                                                              1. Aliasing Errors
                                                                                                                1. Dealiasing Techniques
                                                                                                                2. Applications in Geosciences
                                                                                                                  1. Global Atmospheric Models
                                                                                                                    1. Ocean Circulation
                                                                                                                      1. Seismic Wave Propagation
                                                                                                                  2. Solving Governing Equations
                                                                                                                    1. Ordinary Differential Equations
                                                                                                                      1. Initial Value Problems
                                                                                                                        1. Euler Method
                                                                                                                          1. Forward Euler
                                                                                                                            1. Backward Euler
                                                                                                                              1. Error Analysis
                                                                                                                              2. Runge-Kutta Methods
                                                                                                                                1. Second-Order RK
                                                                                                                                  1. Fourth-Order RK
                                                                                                                                    1. Adaptive Step Size
                                                                                                                                  2. Boundary Value Problems
                                                                                                                                    1. Shooting Method
                                                                                                                                      1. Linear Problems
                                                                                                                                        1. Nonlinear Problems
                                                                                                                                          1. Multiple Shooting
                                                                                                                                          2. Finite Difference Approach
                                                                                                                                            1. Discretization
                                                                                                                                              1. Matrix Formulation
                                                                                                                                                1. Solution Techniques
                                                                                                                                              2. Time-Stepping Schemes
                                                                                                                                                1. Explicit Methods
                                                                                                                                                  1. Stability Constraints
                                                                                                                                                    1. Implementation
                                                                                                                                                      1. Computational Cost
                                                                                                                                                      2. Implicit Methods
                                                                                                                                                        1. Unconditional Stability
                                                                                                                                                          1. Nonlinear Solvers
                                                                                                                                                            1. Computational Efficiency
                                                                                                                                                            2. Stability Analysis
                                                                                                                                                              1. Linear Stability
                                                                                                                                                                1. Nonlinear Stability
                                                                                                                                                                  1. Energy Methods
                                                                                                                                                                  2. CFL Condition
                                                                                                                                                                    1. Derivation
                                                                                                                                                                      1. Physical Interpretation
                                                                                                                                                                        1. Practical Applications
                                                                                                                                                                    2. Partial Differential Equations
                                                                                                                                                                      1. Elliptic Equations
                                                                                                                                                                        1. Laplace and Poisson Equations
                                                                                                                                                                          1. Physical Interpretation
                                                                                                                                                                            1. Boundary Conditions
                                                                                                                                                                              1. Green's Functions
                                                                                                                                                                              2. Solution Techniques
                                                                                                                                                                                1. Direct Methods
                                                                                                                                                                                  1. Iterative Methods
                                                                                                                                                                                    1. Multigrid Methods
                                                                                                                                                                                  2. Parabolic Equations
                                                                                                                                                                                    1. Heat Equation
                                                                                                                                                                                      1. Physical Derivation
                                                                                                                                                                                        1. Analytical Solutions
                                                                                                                                                                                          1. Numerical Solutions
                                                                                                                                                                                          2. Time Integration Methods
                                                                                                                                                                                            1. Explicit Schemes
                                                                                                                                                                                              1. Implicit Schemes
                                                                                                                                                                                                1. Crank-Nicolson Method
                                                                                                                                                                                              2. Hyperbolic Equations
                                                                                                                                                                                                1. Advection and Wave Equations
                                                                                                                                                                                                  1. Characteristics Method
                                                                                                                                                                                                    1. Wave Propagation
                                                                                                                                                                                                      1. Boundary Conditions
                                                                                                                                                                                                      2. Upwind and Lax-Wendroff Schemes
                                                                                                                                                                                                        1. First-Order Upwind
                                                                                                                                                                                                          1. Lax-Wendroff Method
                                                                                                                                                                                                            1. TVD Schemes
                                                                                                                                                                                                      3. Numerical Linear Algebra
                                                                                                                                                                                                        1. Direct Solvers
                                                                                                                                                                                                          1. Gaussian Elimination
                                                                                                                                                                                                            1. Partial Pivoting
                                                                                                                                                                                                              1. Complete Pivoting
                                                                                                                                                                                                                1. Computational Complexity
                                                                                                                                                                                                                2. LU and Cholesky Decomposition
                                                                                                                                                                                                                  1. LU Factorization
                                                                                                                                                                                                                    1. Cholesky Factorization
                                                                                                                                                                                                                      1. Forward and Back Substitution
                                                                                                                                                                                                                    2. Iterative Solvers
                                                                                                                                                                                                                      1. Jacobi and Gauss-Seidel Methods
                                                                                                                                                                                                                        1. Algorithm Description
                                                                                                                                                                                                                          1. Convergence Analysis
                                                                                                                                                                                                                            1. Relaxation Parameters
                                                                                                                                                                                                                            2. Conjugate Gradient Method
                                                                                                                                                                                                                              1. Algorithm Derivation
                                                                                                                                                                                                                                1. Preconditioning
                                                                                                                                                                                                                                  1. Convergence Properties
                                                                                                                                                                                                                                  2. Multigrid Methods
                                                                                                                                                                                                                                    1. Geometric Multigrid
                                                                                                                                                                                                                                      1. Algebraic Multigrid
                                                                                                                                                                                                                                        1. V-cycle and W-cycle
                                                                                                                                                                                                                                    2. Optimization Techniques
                                                                                                                                                                                                                                      1. Gradient-Based Methods
                                                                                                                                                                                                                                        1. Steepest Descent
                                                                                                                                                                                                                                          1. Algorithm Description
                                                                                                                                                                                                                                            1. Convergence Rate
                                                                                                                                                                                                                                            2. Newton's Method
                                                                                                                                                                                                                                              1. Second-Order Information
                                                                                                                                                                                                                                                1. Quasi-Newton Methods
                                                                                                                                                                                                                                                  1. Trust Region Methods
                                                                                                                                                                                                                                                2. Global Optimization Algorithms
                                                                                                                                                                                                                                                  1. Genetic Algorithms
                                                                                                                                                                                                                                                    1. Selection Mechanisms
                                                                                                                                                                                                                                                      1. Crossover and Mutation
                                                                                                                                                                                                                                                        1. Population Dynamics
                                                                                                                                                                                                                                                        2. Simulated Annealing
                                                                                                                                                                                                                                                          1. Cooling Schedules
                                                                                                                                                                                                                                                            1. Acceptance Criteria
                                                                                                                                                                                                                                                              1. Parameter Tuning

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