Field Methods

Numerical methods for fields, partial differential equations, and boundary conditions.

General idea of the topic

Field methods solve the partial differential equations (PDEs) governing continuum physics: electrostatics (Poisson and Laplace), thermal diffusion (Fourier), wave propagation (Maxwell and wave equations), and quantum dynamics (time-dependent Schrödinger equation). Their foundation is space-time discretization onto computational grids where differential operators transform into algebraic matrix systems.

This node covers static relaxation schemes (Gauss-Seidel, SOR), explicit and implicit parabolic solvers (Crank-Nicolson), and Finite-Difference Time-Domain (FDTD) wave propagation.

What you will truly understand

You will understand the Courant-Friedrichs-Lewy stability criterion ($c \Delta t \le \Delta x$), why explicit diffusion schemes suffer from restrictive time steps ($\Delta t < \Delta x^2 / 2D$), and how unitary/implicit schemes ensure unconditional stability.

You will master physical boundary conditions (Dirichlet, Neumann, periodic, and absorbing boundaries).

Recommended learning route

1. Elliptic equations (Poisson/Laplace): 2D finite difference stencils, successive over-relaxation, and electrostatic shielding potentials.
2. Parabolic equations (Diffusion): 1D/2D heat equation, FTCS vs Crank-Nicolson schemes, and heat flux conservation.
3. Hyperbolic equations (Waves): elastic and electromagnetic pulse propagation, reflections, and numerical dispersion.
4. Time-dependent Schrödinger equation: wave-packet scattering, barrier tunneling, and norm conservation.

Physical criterion you must acquire

Numerical causality: the computational domain of dependence must encompass the physical domain of dependence to prevent catastrophic numerical instability.

Connection with real problems

• FDTD simulation of radar antennas, photonic crystals, and optical waveguides.
• Thermal management and heat dissipation in semiconductor chips.
• Quantum tunneling and wave-packet dynamics in nanoscale semiconductors.

Final goal of the topic

Master spatial and temporal discretization to solve Poisson, diffusion, wave, and Schrödinger PDEs in 1D and 2D.