Multidimensional Forward-in-Time and Upstream-in-Space-Based Differencing for Fluids

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Abstract
Multidimensional advection schemes based on the forward-upstream discretization are presented that with only one corrective step produce solutions comparable to the most accurate solutions produced by the multidimensional positive definite advection transport algorithm (MPDATA) family of schemes. The proposed schemes are not positive definite by structure, in contrast to the family of MPDATA schemes. A monotonicity-preserving algorithm is therefore an integral part of the schemes. Based on linear von Neumann analysis and numerical advection experiments in uniform, rotational, and deformational flows, it has been shown that all of the monotone versions of the schemes are stable for |αI| ≤ 0.5, where αI and M are the advective Courant number and the dimensionality of the problem, respectively. Five of the proposed schemes have an amplification error close to, or slightly less than, that of the most accurate versions of the MPDATA scheme. The monotone second-order version of the most accurate scheme is 60% more expensive than the basic second-order MPDATA scheme with one antidiffusive correction step, but 70% cheaper than the corresponding monotone version of MPDATA. In addition, the most accurate of the proposed schemes is more cost efficient than any of the MPDATA schemes. All of the second-order versions of the schemes have a phase error similar to the first-order forward-upstream scheme. The phase error can be reduced by compensating for the second-order forward-upstream discretization error term. If the uniform version of the second-order forward-upstream discretization error term is applied to the schemes, the most accurate scheme becomes up to five times as efficient as the most accurate MPDATA scheme.

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