References: MPDATA

early works:

• Courant et al. 1952 (Comm. Pure Appl. Math 5)
  "On the solution of nonlinear hyperbolic differential equations by finite differences"

general references:

• Richtmyer 1962 (NCAR Technical Notes 63-2)
  "A Survey of Difference Methods for Non-Steady Fluid Dynamics"

MPDATA reviews:

• Smolarkiewicz and Margolin 1998
• Smolarkiewicz and Margolin 2001
• Smolarkiewicz 2006

upwind, MPDATA and other transport solvers for condensational growth:

• Kovetz & Olund 1969 (J. Atmos. Sci. 26)
  "The Effect of Coalescence and Condensation on Rain Formation in a Cloud of Finite Vertical Extent" (Sect. 3.3)
• Tsang & Brock 1982 (Aerosol Sci. Tech. 2)
  "Simulation of Condensation Aerosol Growth by Condensation and Evaporation"
• Smolarkiewicz 1984 (J. Comp. Phys. 54)
  "A fully multidimensional positive definite advection transport algorithm with small implicit diffusion" (Sect. 5.1)
• Tsang & Korgaonkar 1987 (Aerosol. Sci. Tech. (7))
  "Effect of Evaporation on the Extinction Coefficient of an Aerosol Cloud"
• Tsang & Rao 1988 (Aerosol. Sci. Tech. (9))
  "Comparison of Different Numerical Schemes for Condensational Growth of Aerosols"
• Williams & Loyalka 1991
  "Aerosol Science: Theory and Practice" (Sect. 5.19)
• Kostoglou & Karabelas 1995 (Chem. Eng. Comm. 136)
  "Evaluation of Numerical Methods for Simulating an Evolving Particle Size Distribution in Growth Processes"
• Liu et al. 1997 (J. Atmos. Sci.)
  "Variational Optimization Method for Calculation of Cloud Drop Growth in an Eulerian Drop-Size Framework"
• Li et al. 2017
  "Eulerian and Lagrangian approaches to multidimensional condensation and collection" (Appendix A: Upwinding Scheme for a Nonuniform Mesh)
• Morrison et al. 2018 (J. Atmos. Sci. 75)
  "Broadening of Modeled Cloud Droplet Spectra Using Bin Microphysics in an Eulerian Spatial Domain"
• Wei et al. 2020 (MDPI Atmosphere)
  "A Lagrangian Advection Scheme for Solving Cloud Droplet Diffusion Growth"

advection-diffusion as advection-only transport:

• Lange 1978 (J. Appl. Meteorol. 17)
  "ADPIC—A Three-Dimensional Particle-in-Cell Model for the Dispersal of Atmospheric Pollutants and its Comparison to Regional Tracer Studies"
• Smolarkiewicz & Clark 1986 (J. Comp. Phys. 67)
  "The multidimensional positive definite advection transport algorithm: Further development and applications"
• Boybeyi 1990 (SJSU MSc thesis)
  "Development and evaluation of an adaptive volume puff model"
• Yeh, G.-T. (2000)
  "Numerical Methods for Advection-Dominant Transport. Computational Subsurface Hydrology"
• Smolarkiewicz & Szmelter 2005 (J. Comp. Phys. 206)
  "MPDATA: An edge-based unstructured-grid formulation"
• Sousa 2009 (Int. J. Num. Meth. Fluids 61)
  "Insights on a sign-preserving numerical method for the advection–diffusion equation"
• Cristiani 2015 (J. Coupled Syst. Multiscale Dyn. 3)
  "Blending Brownian motion and heat equation"
• Arabas & Farhat 2019 (J. Comp. Appl. Math.)
  "Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations"

MPDATA variants: infinite gauge:

• ...
• Margolin & Shashkov 2006 (Num. Meth. Fluids 50)
  "MPDATA: gauge transformations, limiters and monotonicity"
• Szmelter & Gillard 2020
  "A Multidimensional Positive Definite Remapping Algorithm for Unstructured Meshes"

MPDATA variants: summed-series antidiffusive velocities

• Beason & Margolin 1988 (LLNL report UCRL-99731)
  "DPDC: A second-order monotone scheme for advection"
• Margolin & Shashkov 2006 (Num. Meth. Fluids 50)
  "MPDATA: gauge transformations, limiters and monotonicity"

MPDATA on a sphere:

• Smolarkiewicz & Rasch 1991 (J. Atmos. Sci. 48)
  "Monotone Advection on the Sphere: An Eulerian Versus Semi-Lagrangian Approach"
• Hundsdorfer et al. 1995
  "A Positive Finite-Difference Advection Scheme"
• Margolin, Shashkov & Smolarkiewicz 2000 (Comp. Meth. Appl. Mech. Eng. 187)
  "A discrete operator calculus for finite difference approximations"
• Jaruga et al. 2015 (Geosci. Model Dev.)
  "libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations"
• Waruszewski et al. 2018
  "MPDATA: Third-order accuracy for variable flows"
• Ackmann et al. 2020
  "Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows"

MPDATA performance:

• Smolarkiewicz & Rasch 1991 (J. Atmos. Sci. 48)
  "Monotone Advection on the Sphere: An Eulerian Versus Semi-Lagrangian Approach"
• Margolin, Shashkov & Smolarkiewicz 2000 (Comp. Meth. Appl. Mech. Eng. 187)
  "A discrete operator calculus for finite difference approximations"
• Smolarkiewicz and Szmelter 2005 (J. Comp. Phys. 206)
  "_MPDATA: an edge-based unstructured-grid formulation"

method of lines/characteristics and advection eq:

• Hundsdorfer & Trompert 1994 (Appl. Num. Math. 13)
  "Method of lines and direct discretization: a comparison for linear advection"
• Sarra 2003 (JOMA)
  "The Method of Characteristics & Conservation Laws"

upwind/upstream/donor-cell/...

• Leonard 1979 (Computer Methods in Applied Mechanics and Engineering 19)
  "A stable and accurate convective modelling procedure based on quadratic upstream interpolation"

modified equation analysis

• Warming & Hyett 1974 (J. Comp. Phys. 14)
  "The modified equation approach to the stability and accuracy analysis of finite-difference methods"
• Trangenstein 2009 (ISBN: 9780521877275)
  "Numerical Solution of Hyperbolic Partial Differential Equations"