From the summary: The present paper reports constrained optimization of explicit Runge-Kutta (RK) schemes, coupled with optimal upwind compact scheme to achieve dispersion relation preservation (DRP) property for high performance computing. Essential ideas of optimization employed in arriving at the proposed time integration scheme are extension of the earlier work reported in M. K. Rajpoot et al. [J. Comput. Phys. 229, No. 10, 3623–3651 (2010; Zbl 1190.65139)]. This is in turn an application of the correct error evolution equation in T. K. Sengupta et al. [J. Comput. Phys. 226, No. 2, 1211–1218 (2007; Zbl 1125.65337)]. Resultant DRP scheme demonstrated the idea for explicit spatial central difference schemes. Present work is similar, extending it for near-spectral accuracy compact schemes. Practical utility of the developed method is demonstrated by solution of model problems and for flow problems by solving Navier-Stokes equation, some of which cannot be solved by conventional schemes, as the problem of rotary oscillation of cylinder.
Developed method is calibrated with: (i) flow past a circular cylinder performing rotary oscillation at \(Re = 150\) and (ii) flow inside a 2D lid-driven cavity (LDC) at Reynolds numbers of \(Re = 1000\) and \(Re = 10,000\). Quantitative and qualitative comparisons show excellent match for rotary oscillation cylinder cases with experimental results. Results for LDC for \(Re = 1000\) and results for \(Re = 10,000\) are compared with recent published ones showing triangular vortex in the core.