Upstream Nonoscillatory Advection Schemes

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Abstract
Upstream nonoscillatory (UNO) advection schemes are derived by optimizing existing classical advection schemes and combining them in different monotonic zones to avoid flux limiters for simplicity. The UNO schemes are extended to irregular grids in the form of upstream midflux linear interpolation with symmetrical gradients and are adapted to multidimensions with an advective–conservative operator. They are given in finite-volume flux form and hence are consistent and conservative. They also preserve constancy and linear correlation. Implicit numerical diffusivity of these schemes is also derived and used as a guideline for the selection of advection schemes. One- and two-dimesional tests are used for comparisons with their classical counterparts. Multiple-cell grids are used to test the irregular grid formulation and demonstrate their performance. The simple second-order UNO2 scheme may be accurate enough when the physical diffusion or numerical smoothing term is larger than the numerical diffusion. The third-order UNO3 scheme has very small self-constrained numerical diffusion and is suitable for general atmospheric and oceanic tracer advection.

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