🧮 Numerical Methods for PDEs

Learn the fundamentals of finite difference methods for discretizing partial differential equations into difference equations. Derive FTCS (Forward Time Central Space) and BTCS (Backward Time Central Space) methods, and implement stability evaluation using von Neumann stability analysis and CFL condition. Applications to heat conduction equations are also covered.

Learn the Crank-Nicolson method that combines the advantages of implicit and explicit methods. Prove unconditional stability and theoretical accuracy evaluation, and implement efficient solution of 2D heat equations using Alternating Direction Implicit (ADI) method and treatment of boundary conditions.

Learn the finite element method that can handle complex domains and boundary conditions. From the variational principle of Galerkin method to construction of shape functions, element division, and assembly of stiffness matrices, and solve 1D and 2D Poisson equations using Python.

Learn high-precision spectral methods (Fourier method, Chebyshev method, pseudo-spectral method) and probabilistic approach of Kinetic Monte Carlo method. Handle periodic and non-periodic boundary conditions, and implement applications to diffusion equations and reaction-diffusion systems.

Apply numerical methods for PDEs to materials processes. Heat treatment simulation, solidification process analysis using Phase-Field models, and diffusion simulation (Fick equation) among other practical materials process simulations are implemented using Python.