TVDal: Total variation diminishing scheme with alternating limiters to balance numerical compression and diffusion

Abstract

Total variation diminishing (TVD) advection schemes are widely used in ocean modelling. Due to the constraints of flux limiters, TVD schemes with common single flux limiters of second-order or third-order accuracy exhibit defects associated with numerical compression or diffusion. To reduce the numerical errors induced by these numerical defects, a TVD method employing alternating flux limiters (TVDal) is proposed. The principle of TVDal is to alternately use a compressive and a diffusive limiter in different time steps. Through balancing the compression of the Superbee limiter and the diffusion of other flux limiters, a reduction of ∼60% of numerical errors in one-dimensional benchmark tests can be achieved with the TVDal scheme. In addition, the TVDal scheme combined with the Strang-splitting method is proposed for multi-dimensional advection. For an idealized 2D experiment with oblique advection, the Strang-splitting TVDal scheme features good shape retention with much lower numerical errors. The TVDal scheme has also been applied to model tidally induced internal lee waves, indicating that TVDal has the potential ability to balance the numerical compression and diffusion of conventional TVD schemes for cases with highly nonlinear, stratified, and nonhomogeneous flow fields. Finally, the principles of the TVDal scheme also show the potential application to conventional high-order TVD schemes.