An Enhanced Symmetrized Implicit Trapezoidal Rule for Dampening the Oscillatory Behaviour of Numerical Solutions

The symmetrized implicit trapezoidal rule (SITR) is a novel finite difference method that involves symmetrization of the calculated numerical solutions using the implicit trapezoidal rule (ITR). By solving differential equations using large space and time steps, compared with conventional methods, the SITR method can generate more accurate results by preventing oscillations in the numerical solutions. In this study, the effect of parameters such as the space step and time step on the accuracy and computational efficiency of the active one-step symmetrized implicit trapezoidal rule (1SITR), the active two-step symmetrized implicit trapezoidal rule (2SITR) and the generalized trapezoidal differencing time-marching (GTDTM) method is investigated in solving aeroacoustics problem involving a linear advection equation. The results of the study show that compared to the conventional GTDTM method, both the enhanced 1SITR and 2SITR methods possess damping characteristics that reduce the oscillatory behaviour of the numerical solution, leading to better solution accuracy at large spatial and temporal step sizes. At smaller temporal step sizes, the increase in the dissipative characteristics of 1SITR and 2SITR reduces their accuracy. The computational time for both 1SITR and 2SITR is greater than that for GTDTM due to the extra symmetrization calculations. This study presents a unique contribution through the development of two symmetrized implicit trapezoidal rule methods which provide improved damping characteristics, allowing for more stable and accurate solutions even under coarse discretization settings.