Evaluating Numerical Stability in High-Accuracy Simulations: A Comparative Study of Time Discretization Methods for the Linear Convection Equation

Abstract

With the advent of technology, it has become possible to perform direct numerical simulations and the demand for high accuracy computing is increasing. Numerical simulations play an important part in understanding physics of the flow and instability mechanism in flows. For high accuracy, numerical schemes must be chosen that satisfy the physical dispersion relation, should not amplify or attenuate the solution and resolve all possible length and time scales. In the present paper, spectral stability analysis of linear convection equation is performed using first order forward difference (FD1) method and fourth order Runge Kutta (RK4) method, consisting of four stages, for time discretization and a second order central difference (CD2) method for evaluating spatial derivative. The results show that the presence of numerical instability for FD1 method is independent of the CFL number, consistent with the stability analysis which showed FD1 method to be unconditionally unstable. However, for RK4 method, the solution is found to be neutrally stable only for a particular range of CFL number, even stable solution introduced error by attenuating the computed or analytical solution.

Keywords

• spectral stability
• convection equation
• rk4
• fd1
• cd2