MPDATA: An edge-based unstructured-grid formulation

  • Authors:

  • Piotr K. Smolarkiewicz;Joanna Szmelter

  • Affiliations:

  • Mesoscale and Microscale Meteorology Division, National Center for Atmospheric Research, P.O. BOX 3000, Boulder, CO 83307-3000, USA;Royal Military College of Science, Cranfield University, Shrivenham, Swindon SN6 8LA, UK

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2005

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Abstract

We present an advancement in the evolution of MPDATA (multidimensional positive definite advection transport algorithm). Over the last two decades, MPDATA has proven successful in applications using single-block structured cuboidal meshes (viz. Cartesian meshes), while employing homeomorphic mappings to accommodate time-dependent curvilinear domains. Motivated by the strengths of the Cartesian-mesh MPDATA, we develop a new formulation in an arbitrary finite-volume framework with a fully unstructured polyhedral hybrid mesh. In MPDATA, as in any Taylor-series based integration method for PDE, the choice of data structure has a pronounced impact on the technical details of the algorithm. Aiming at a broad range of applications with a large number of control-volume cells, we select a general, compact and computationally efficient, edge-based data structure. This facilitates the use of MPDATA for problems involving complex geometries and/or inhomogeneous anisotropic flows where mesh adaptivity is advantageous. In this paper, we describe the theory and implementation of the basic finite-volume MPDATA, and document extensions important for applications: a fully monotone scheme, diffusion scheme, and generalization to complete flow solvers. Theoretical discussions are illustrated with benchmark results in two and three spatial dimensions.