Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings - Md Joni Alam; Ahmed Ramady; M. S. Abbas; K. El-Rashidy; Md Tauhedul Azam; M. Mamun Miah (AppliedMath)
The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the <semantics>L1</semantics> norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme.