Upstream Non-Oscillatory Advection Schemes Suitable for Ocean Wave Models

The first order upstream advection scheme used in the WAM ocean wave models is known too diffusive and is responsible for the underestimation of long distance swells. High order advection schemes used in other wave models, like the 3 order ULTIMATE QUICKEST scheme in WAVEWATCH III, are 'too good' as it does not smooth out the garden sprinkler effect. The purpose of this paper is to find a suitable 2 order scheme, which reduces the 1 order diffusion but keeps enough smoothing for the discrete direction and spectral bins. Of course, it should be free from oscillations which haunt many high order schemes. The proposed upstream nonoscillatory 2 order (UNO2) scheme is adapted from the MINMOD scheme and simplified to favour vectorization on super computers. The scheme is extended to 2-D with the multidimensional advective-conservative hybrid operator (MACHO), which keeps the 2-D scheme stable under the 1-D CFL condition and eliminates time-splitting errors. The scheme has been tested in the Met Office global and regional wave models. It is more effective than the 3 order Gadd scheme used in the Met Office operational wave models and even shows better results than the old scheme. A 3 order scheme (UNO3), similar to the 3 order ULTIMATE QUICKEST scheme but replaced its flux limiter part by the UNO2 scheme, is also presented. The UNO3 scheme is of the same accuracy as the ULTIMATE QUICKEST scheme but more efficient. Two more intermediate schemes (UNO2 and UNO3) are derived from the van Leer homonic average. Classical numerical (1-D constant speed and 2-D solid rotation and deformation) tests have demonstrated that these schemes are non-oscillatory, conservative and shape-preserving. These schemes, especially the UNO2 scheme, are recommended for replacing the 1 order upstream scheme in WAM models.