Abstract
Multidimensional positive definite advection transport algorithm (MPDATA) was proposed in the early eighties as a simple positive‐definite advection scheme with small implicit diffusion, for evaluating the advection of water‐substance constituents in atmospheric cloud models. Over the two decades, MPDATA evolved from an advection scheme into a class of generalized transport algorithms that expand beyond advective transport to alternate PDEs and complete fluid models with a wide range of underlying governing equations. Recently, MPDATA has attracted attention in the context of several mutually‐beneficial developments such as (i) quantification of MPDATA implicit turbulence modelling capability in the spirit of monotonically integrated large eddy simulations (MILES), (ii) extensions to flow solvers cast in generalized time‐dependent curvilinear coordinates, and (iii) unstructured‐grid formulations. The aim of this paper is to assist the special issue on MPDATA methods for fluids by providing an up to date comprehensive review of the approach, including the underlying concepts, principles of implementation, and guidance to the accumulated literature. Copyright © 2005 John Wiley & Sons, Ltd.