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Volume 17, No. 1, January 2017, Pages 278-289 PDF(3.61 MB)  
doi: 10.4209/aaqr.2016.06.0271   

Backward Integration of Diffusion Equation

Wen-Yih Sun1,2,3, 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


  • 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.



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