Numerical Analysis of Third Order Advection-Viscous Wave Equation Using Finite Difference Method - Olima, Samuel Owiti (JOOUST, 2023)
Wave equation is a linear hyperbolic partial differential equation (PDE) which describes the propagation of a variety of waves arising in physical situations. In its simplest form, the one dimension wave equation refers to a scalar function u = u(x, t), which satisfy the PDE utt = C2uxx. When the wave propagates in complex media, form of the governing wave equation changes, so in particular a viscous loss in term µuxxt and the advection term αux are added to the right hand side of the equation thus: utt = C2uxx+µuxxt+αux. Many researchers have tried to solve this equation analytically and numerically without advection component using laplace transform or finite difference method. In our study we solved viscous wave equation with advection component included. The numerical approach we have used is finite difference method to discretize the equation, its associated initial and boundary conditions by their finite difference analogue. We managed to analyze third order advection viscous eave equations by developing two numerical schemes namely Central Crank Nicolson and Forward Crank Nicolson Difference Schemes analyzed their stability and consistency to ascertain convergence of the schemes to unique solutions of the original equation. We used MATLAB to find the solutions of matrices developed from the algebraic equations and to draw tables and graphs. The trend of graphs shows that the amplitude of wave decreases with increase in distance. We hope that the solution we obtained may be useful in the analysis and prediction of viscous advection wave equation problems which have great application in sound instruments, construction engineering and medicine. 1.544.
Keywords: Numerical Analysis.; Advection-Viscous Wave Equation.; Finite Difference Method.