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

Abstract

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 ΣMI |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 sc...