On the application of the two-time stepping Euler forward Runge-Kutta schemes to the shallow water equations: Global truncation error, numerical viscosity, consistency, energy conservation, inertial stability and phase error Ocean Modelling

This study aims to investigate several issues relating to the application of Runge-Kutta (RK) schemes, which are popularly applied to single-variable ordinary differential equations (ODEs) and multi-variable partial differential equations (PDEs), such as shallow water equations (SWEs). Starting with the original mathematical definition of the RK scheme, it was found that RK schemes are fundamentally similar to the first-order Euler forward predictor-corrector (PC) scheme. It is shown that 2-time stepping RK and other schemes actually are the PC schemes. Truncation error analysis shows that the 2-, 3-, and 4-stage PC/RK (or PC2/RK2, PC3/RK3, and PC4/RK4, respectively) are of first-order accuracy in time due to the nature of the first-order Euler forward scheme at each stage. Truncation analysis shows that 1st-order PC/RK schemes discard the physically-based bi-harmonic viscosity term. This is equivalent to adding the same, but a negative viscosity to the numerical scheme, and is inconsistent with their original PDEs if compared to the 2nd-order accurate leapfrog scheme. It is shown that each step of the multi-stage PC (i.e., RK) iterative process improves the precision of its previous stage in terms of amplification factor and phase speed. Using a pure advection system, it is proven that any 1st-order Euler forward scheme in time and space cannot conserve energy. It is also confirmed that the explicit treatment of unweighted and equally-wighted Coriolis terms in PC/RK schemes produce inertial instability, which must be dampened using numerical filters or by adding unrealistically high viscosity. As a result, models that use such schemes are overly-damping, leading to smoothing of important dynamical processes, such as mesoscale eddies, vertical stratification, and the strength of horizontal fronts. Three types of 3-stepping (at n-1, n, n + 1, leapfrog-like) schemes are investigated including the leapfrog-trapezoidal scheme, Adams-Bashforth scheme, and leapfrog-hoRA (high order Robert-Asselin) filter scheme. It is found that the leapfrog-hoRA scheme is advantageous in that 1) it is of second-order accuracy in general, and can be configured to achieve third-order accuracy when β=0.4, 2) it introduces no numerical viscosity, and 3) it produces nearly neutral inertial stability in terms of both amplification factor and phase speed.