Courant-Friedrichs-Lewy Condition for Analysis of Convergence and Stability of Explicit Forward Time Central Space Scheme for Three-Dimensional Wave Equation

The aim of this research is to examine Courant-Friedrichs-Lewy condition for the analysis of convergence and stability of explicit forward time central space scheme for a three-dimensional wave equation. The wave equation, which models physical phenomena such as sound and electromagnetic wave propagation, is discretized using finite difference methods in both time and space. A central difference scheme is implemented to approximate the second-order derivatives across spatial and temporal domains. The CFL condition is derived as a criterion to ensure numerical stability and is shown to depend on the wave speed and spatial grid resolution. The explicit update scheme is constructed and analyzed under uniform grid spacing. Through von Neumann stability analysis, the amplification factor is expressed using Fourier modes and Euler’s identity. The characteristic equation for the scheme is derived, and its roots are examined to determine the conditions for numerical stability. The CFL number $\lambda =\frac{c\text{Δ}t}{h}$ is introduced and bounded to prevent error magnification. The analysis confirms that for stability, the time step must satisfy $\text{Δ}t\le \frac{h}{c\sqrt{3}}$ . Convergence is discussed in the context of satisfying both consistency and stability criteria. The initial and boundary conditions necessary for realistic modeling are incorporated. This work validates that adherence to the CFL condition is essential for reliable and accurate simulation of three-dimensional wave propagation using explicit finite difference methods.