MULTIDIMENSIONAL FORWARD-IN-TIME AND UPSTREAM-IN-SPACE-BASED DIFFERENCING FOR FLUIDS

Multidimensional advection schemes based on the forward-upstream discr etization are presented that with only one corrective step produce sol utions comparable to the most accurate solutions produced by the multi dimensional positive definite advection transport algorithm (MPDATA) f amily of schemes. The proposed schemes are not positive definite by st ructure, in contrast to the family of MPDATA schemes. A monotonicity-p reserving algorithm is therefore an integral parr of the schemes. Base d on linear von Neumann analysis and numerical advection experiments i n uniform, rotational, and deformational flows, it has been shown that all of the monotone versions of the schemes are stable for Sigma(l)(M )\alpha'\ less than or equal to 0.5, where alpha' and M are the advect ive Courant number and the dimensionality of the problem, respectively . Five of the proposed schemes have an amplification error close to, o r 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 correspond ing 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 s imilar to the first-order forward-upstream scheme. The phase error can be reduced by compensating for the second-order forward-upstream disc retization error term. If the uniform version of the second-order forw ard-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.