Solving first-order systems of linear hyperbolic partial differential equations using fast Fourier transform

In this paper, we address the exact solution of the general first-order systems of linear hyperbolic partial differential equations using the Fourier transformation technique. This transformation converts the system from the physical domain into a system of first-order ordinary differential equations in the frequency domain. Utilizing this method, we then solve the wave equation system in n dimensions. Following the derivation of the exact solution, we introduce a numerical algorithm based on the fast Fourier transform (FFT) to solve the same system numerically. This approach leverages the exact solution obtained in the frequency domain, facilitating efficient and precise numerical computations. To validate the effectiveness of the proposed numerical algorithm, we present numerical experiments. These experiments demonstrate the precision and efficiency of the algorithm when applied to the wave equation system in both two and three dimensions, considering scenarios with continuous and discontinuous initial data.