Wave properties of fourth-order fully implicit Runge–Kutta time integration schemes

• Global spectral analysis of IRK2 time integration scheme performed.
• Numerical properties of IRK2 quantified at all nodal points in (Nc, kh)-plane.
• Results indicate the use of refined grids at high Nc for better DRP properties.
• 1D convection equation solved numerically to further explain this aspect.
• Conclusions drawn are equally applicable to any implicit time-integration scheme.

A global spectral analysis of fully implicit, two-stage, fourth-order Runge–Kutta method (IRK2) is presented here. The essential numerical properties of the IRK2 time-integration scheme, viz., nodal amplification factor, phase speed and group velocity have been quantified and plotted at all the nodal points of the computational domain as functions of the CFL (Courant–Friedrichs–Lewy number) and wavenumber for various explicit and implicit spatial discretization schemes. Results show that IRK2 scheme produces neutral stability at all interior nodes for all CFL numbers. This has been shown that while using IRK2 scheme one has to use refined grid at high CFL numbers for dispersion relation preservation property.