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FM Dojo > Partial Differential Equations and Boundary Value Problems > Chapter 5
🎯 Learning Objectives
- Understand the fundamentals of finite difference method (FDM) and various schemes
- Master the theory and implementation of finite element method (FEM)
- Learn the characteristics of time integration schemes (explicit and implicit methods)
- Understand the theoretical foundations of stability analysis and convergence
- Learn mesh generation and element selection
- Understand extensions to 2D and 3D problems
- Master efficient handling of sparse matrices
- Implement applications to process simulations (heat treatment, reaction-diffusion)
📖 Fundamentals of Numerical Methods
Classification of Numerical Methods
Finite Difference Method (FDM) :
- Replaces derivatives with difference approximations
- Easy to implement on structured grids
- Difficult to apply to complex geometries
Finite Element Method (FEM) :
- Uses weak formulation based on variational principles
- Handles complex geometries with unstructured grids
- High-accuracy interpolation within elements
Finite Volume Method (FVM) :
- Handles conservation laws in integral form
- Widely used in fluid dynamics
- Strictly conserves mass and energy
Stability and Convergence
Stability : Condition that numerical errors do not diverge during time evolution
CFL Condition (Courant-Friedrichs-Lewy) : Stability condition for wave equations
\[ C = c \frac{\Delta t}{\Delta x} \leq C_{\text{max}} \]
Convergence : Property of approaching the true solution as mesh width \(\Delta x \to 0\)
Consistency : Property that the difference scheme converges to the differential equation
Lax Equivalence Theorem : Consistency + Stability ⇒ Convergence
Summary
- The finite difference method is easy to implement but difficult to apply to complex geometries. Understanding the characteristics of FTCS, BTCS, and Crank-Nicolson schemes is important
- The finite element method is based on variational principles and handles complex geometries with unstructured grids. Linear triangular elements are fundamental
- Stability and convergence are fundamental concepts that ensure the reliability of numerical methods. The CFL condition and Lax equivalence theorem are important
- Adaptive mesh refinement enables efficient high-accuracy solutions
- Time-dependent problems are time-integrated using explicit or implicit methods after semi-discretization
- Nonlinear problems are solved using iterative methods such as Newton-Raphson method
- Efficient handling of sparse matrices is key to large-scale problems
- Rich practical applications to materials science such as process simulations (quenching, thermal stress)
← Chapter 4: Calculus of Variations and Optimization Series TOP
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