Finite Difference Method (FDM) for Field Discretization and Numerical Solution

The Finite Difference Method (FDM) discretizes the computational domain into numerous small grid elements. These grid cells enable more accurate characterization of structural features and variations within the field domain. By decomposing the domain into multiple grid elements, we can precisely compute various parameters across the field. The implementation typically involves defining a grid spacing parameter (Δx, Δy, Δz) and replacing continuous derivatives with finite difference approximations. For Poisson's equation ∇²φ = f, the second derivatives are commonly approximated using central difference schemes: ∂²φ/∂x² ≈ (φ(x+Δx) - 2φ(x) + φ(x-Δx))/Δx². This transformation converts the continuous partial differential equation into a system of algebraic equations solvable through matrix operations. The resulting sparse matrix system can be efficiently solved using iterative methods like Gauss-Seidel or conjugate gradient algorithms. This numerical approach provides high-resolution characterization of field variations and structural properties, leading to deeper insights into underlying physical phenomena.