Two Methods to Discretize the Wave Equation in Time and Space

This study proposes two methods to derive exact discrete-time and space models for the wave equation as an example of a hyperbolic linear partial differential equation. The first method is based on the separation of variables method, an extension of the previous studies, and discretizes time and space separately. This method can accurately discretize when the initial conditions are expressed as a sum of sinusoidal functions. The second method is based on d'Alembert's method, which can be used strictly with arbitrary initial conditions, and discretizes time and space simultaneously. However, this method requires that the space be one-dimensional. Both models can be computed online, and their use is expected to have applications in control system design.