Improvement of accuracy and stability in numerically solving hyperbolic equations by IDO (Interpolated Differential Operator) scheme with Runge-Kutta time integration

In order to solve hyperbolic partial differential equations by means of the Interpolated Differential Operator (IDO) scheme, time integration of the dependent variable has been carried out by Taylor expansion, and time differentiation has been performed by replacing it with a spatial differentiation. However, in such a method, the time accuracy is limited by the order of the interpolation function and in addition the spatial accuracy is not sufficient in multidimensional problems due to the complexity of the calculations. In terms of numerical stability, the stable region indicated by the CFL (Courant-Friedrich-Levy) number is narrow. Hence, in order to improve the space-time accuracy and to secure numerical stability, time integration by the Runge-Kutta method is applied. Further, a method for increasing the order of the Runge-Kutta method without increasing the computational cost is proposed, taking advantage of the characteristics of the IDO scheme with the physical quantity and its spatial derivative as the dependent variables. The two-dimensional advection equation and the one-dimensional wave equation are solved and the results are quantitatively compared with those obtained by the conventional Taylor expansion. Also, in order to demonstrate the adaptability of this approach to practical problems, we consider Williamson's test case 5 for the shallow-water equation in spherical geometry. It is found that the Runge-Kutta method for multiple dimensions yields accuracy and stability higher than those of the Taylor expansion, demonstrating the effectiveness of the approach.