Numerical Methods for Partial Differential Equations
( 25 Modules )

Module #1
Introduction to Partial Differential Equations
Overview of PDEs, types of PDEs, and their applications

Module #2
Mathematical Preliminaries
Review of linear algebra, calculus, and functional analysis

Module #3
Finite Difference Methods
Introduction to finite difference methods for parabolic and hyperbolic PDEs

Module #4
Finite Difference Methods for Parabolic PDEs
Explicit and implicit finite difference schemes for parabolic PDEs

Module #5
Finite Difference Methods for Hyperbolic PDEs
Explicit and implicit finite difference schemes for hyperbolic PDEs

Module #6
Stability Analysis for Finite Difference Methods
von Neumann stability analysis and CFL condition

Module #7
Finite Element Methods
Introduction to finite element methods for elliptic PDEs

Module #8
Finite Element Methods for Elliptic PDEs
Galerkin methods and Ritz methods for elliptic PDEs

Module #9
Finite Element Methods for Parabolic PDEs
Time-stepping schemes for parabolic PDEs using finite elements

Module #10
Finite Element Methods for Hyperbolic PDEs
Discontinuous Galerkin methods for hyperbolic PDEs

Module #11
Error Analysis for Finite Element Methods
A priori and a posteriori error estimates for finite element methods

Module #12
Finite Volume Methods
Introduction to finite volume methods for convection-diffusion PDEs

Module #13
Finite Volume Methods for Convection-Diffusion PDEs
Upwind and central difference schemes for convection-diffusion PDEs

Module #14
Boundary Element Methods
Introduction to boundary element methods for elliptic PDEs

Module #15
Boundary Element Methods for Elliptic PDEs
Direct and indirect boundary element methods for elliptic PDEs

Module #16
Numerical Solution of Nonlinear PDEs
Numerical methods for nonlinear PDEs, including fixed-point iteration and Newtons method

Module #17
Iterative Methods for Linear Systems
Iterative methods for solving linear systems, including Jacobi, Gauss-Seidel, and GMRES

Module #18
Preconditioning Techniques
Preconditioning techniques for iterative methods, including ILU and multigrid

Module #19
Mesh Generation and Adaptation
Mesh generation and adaptation techniques for numerical methods

Module #20
Computational Applications
Computational applications of numerical methods for PDEs, including fluid dynamics and heat transfer

Module #21
Validation and Verification
Validation and verification of numerical methods for PDEs

Module #22
Uncertainty Quantification
Uncertainty quantification techniques for numerical methods for PDEs

Module #23
Parallel Computing
Parallel computing techniques for large-scale numerical simulations

Module #24
Software Packages
Overview of software packages for numerical methods for PDEs, including MATLAB, Python, and FEniCS

Module #25
Course Wrap-Up & Conclusion
Planning next steps in Numerical Methods for Partial Differential Equations career

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Numerical Methods for Partial Differential Equations ( 25 Modules )

Numerical Methods for Partial Differential Equations
( 25 Modules )