Solving the First Order 1-D Wave Equation: A Comparative Study of Upwind and Euler BTCS Methods

Abstract

The simulation of wave propagation in one-dimensional media is fundamental in understanding dynamic systems governed by partial differential equations (PDEs). This study presents a comparative numerical analysis of two finite difference methods, First Upwind Differencing (explicit) and Euler Backward Time Centered Space (BTCS, implicit), applied to the first-order 1-D linear wave equation. The equation models the transport of a disturbance in a closed tube using a constant wave speed of 300 m/s. Both methods are implemented in MATLAB, and simulation results are analysed based on stability, accuracy, and computational characteristics. The upwind scheme demonstrates satisfactory performance with moderate accuracy but suffers from numerical dissipation at lower grid resolutions. In contrast, the Euler BTCS method shows robust stability and higher fidelity to the initial condition, even with relatively large time steps. Results confirm that both methods are capable of generating stable solutions; however, the implicit method offers greater reliability for long-term simulations. This study highlights the trade-offs between computational simplicity and numerical robustness in solving hyperbolic PDEs using finite difference schemes.