📖 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)