Natural Cubic Spline for Hyperbolic equation with constant coefficients

Abstract

In this Paper we have considered second-order hyperbolic equations by implementing Natural cubic spline (NCS) method. Wave propagation and dynamic systems represents hyperbolic equations. We have considered the class of hyperbolic partial differential equations (PDEs) with constant coefficients and implemented NCS method both explicitly and Implicitly. In our implementation we replaced spatial derivatives  by second derivative of Natural cubic spline and time derivatives by central finite difference operator. To show the effectiveness of proposed method we have considered numerical examples having both Dirichlet and Neumann conditions. In order to evaluate the NCS method's performance in managing various boundary behaviours which are frequently seen in real-world applications like heat conduction issues or wave propagation in bounded domains. The results are represented graphically exhibiting the accuracy of proposed NCS method. To check the efficiency of the NCS method we compared the results with analytical solution. The research not only demonstrates the Natural Cubic Spline's flexibility in resolving hyperbolic PDEs, but also emphasizes its benefits, including its capacity to smooth spatial discretization, adjust to different boundary conditions, and work with both explicit and implicit time integration schemes. The findings verify that the NCS approach is a reliable tool for numerical simulations in domains where hyperbolic equations are used, including fluid dynamics, acoustics, and other domains.