Wen-Yih Sun 1,2,3, Oliver M. Sun4

  • 1 Department of Earth, Atmospheric and Planetary Sciences, Purdue University, W. Lafayette, IN 47907, USA
  • 2 Department of Atmospheric Sciences, National Central University, Zhongli, Taoyuan City 32001, Taiwan
  • 3 Hydrospheric Atmospheric Research Center (HyARC), Nagoya University, Nagoya 464-8601, Japan
  • 4 Naval Undersea Warfare Center, Newport, RI 02841-1708, USA

Received: June 28, 2016
Revised: October 30, 2016
Accepted: October 31, 2016
Download Citation: ||https://doi.org/10.4209/aaqr.2016.06.0271  

  • Download: PDF

Cite this article:
Sun, W.Y. and Sun, O.M. (2017). Backward Integration of Diffusion Equation. Aerosol Air Qual. Res. 17: 278-289. https://doi.org/10.4209/aaqr.2016.06.0271


  • Diffusion equation is converted to integral equations of concentration in 5 cells.
  • Diffusion is treated as subgrid-concentration-mass flux across cells’ boundaries.
  • They are successfully integrated in both forward and backward in time with diffusion.
  • Both integrated results very differ from Lagrangian-trajectory method without mixing.
  • They can be used to assess pollutants at sources from downwind region or vice versa.



When the parabolic differential equation is integrated backward in time, it can create unwanted shortwaves with large amplitude. Hence, instead of solving it as a differential equation, the diffusion equation is converted to the equations of volume-integrated-concentration, and mixing/diffusion is treated as subgrid-turbulent-fluxes across the cell boundaries. Those equations become a set of linear algebra equations and can be solved in both forward- and backward-in time. The proposed method has been validated by the numerical simulations of an idealized case, which consists of 5 different sizes of concentric cylinders with different species. The time evolution of compositions shows that the concentrations in each cylinder can change drastically with time. For the data collected at downwind region, the proposed reverse-in-time integration can be used to assess the concentrations at the source regions, which can be quite different from those derived from the conventional backward-trajectory method without mixing. It also shows that the traditional forward-trajectory or backward-trajectory method without mixing (i.e., Lagrangian method) widely used in meteorology and air pollution can misinterpret the property of fluid parcel at both upwind and downwind regions significantly.

Keywords: Forward and backward integration; Diffusion; Parabolic and hyperbolic equations; Mixing/Diffusion-time scale; Turbulence; Pollution

Share this article with your colleagues 


Subscribe to our Newsletter 

Aerosol and Air Quality Research has published over 2,000 peer-reviewed articles. Enter your email address to receive latest updates and research articles to your inbox every second week.

77st percentile
Powered by
   SCImago Journal & Country Rank

2022 Impact Factor: 4.0
5-Year Impact Factor: 3.4

Aerosol and Air Quality Research partners with Publons

CLOCKSS system has permission to ingest, preserve, and serve this Archival Unit
CLOCKSS system has permission to ingest, preserve, and serve this Archival Unit

Aerosol and Air Quality Research (AAQR) is an independently-run non-profit journal that promotes submissions of high-quality research and strives to be one of the leading aerosol and air quality open-access journals in the world. We use cookies on this website to personalize content to improve your user experience and analyze our traffic. By using this site you agree to its use of cookies.