Several approaches to modeling diffusion in combination with other processes are illustrated.


The following approaches are detailed: 

1-D Diffusion modeled as a partial differential equation: Model # 0330 
Partial differential equation with no flux boundary conditions (Neumann), initialized with a centered spike. 

1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation: Model # 0364 
Same as 1-D Diffusion with consumption of metabolite as a function of x. 

1-D diffusion-advection equation with Robin boundary condition: Model # 0169 
Similar to 1-D diffusion with added advection term. Initial value is zero. There is an inflow concentration, Cin. Uses a Robin condition for inflow boundary condition. 

Random Walks of multiple particles in 1 dimension: Model # 0184 
Multiple realizations of 1-D random walks. Final positions are binned and compared to theoretical calculations. 

Random Walk of single particle in 2 dimensions: Model # 0372 
A single random walk in two dimensions is plotted wth with green circles, marked with starts every nsteps. Three kinds of steps may be chosen: 
(1) fixed step sizes 
(2) random step sizes 
(3) random step size with random angle. 

Fractional Brownian Motion Walk in 2 dimensions: Model # 0374 
Uses increments of fractional Gaussian noise to create fractional Brownian Motion using the Davies-Harte algorithm. (See FGP model for details.) Illustrates that steps can be Gaussian, but the Hurst Coefficient measuring correlation of steps is highly important. 

Diffusion in a uniform slab: Model # 176 
Similar to 1-D Diffusion with addition of a partition coefficient lambda, the ratio of the concentration immediately inside the region to that outside. Uses Dirichlet boundary conditions. 

Diffusion in two uniform slabs with different diffusivities: Model 0212 
Similar to Diffusion in a uniform slab, but with the diffusion coefficient containing a discontinuity and the boundary of two media. 

Laplace's equation in 2-D with Dirichlet Boundary conditions: Model # 0363 
Laplace's equation in two dimension is solved using ordinary different equations and solved again used 1-d partial differential equations. both methods use second order accurate finite difference approximations. 



Models Referenced

Key Terms

diffusion tutorial, diffusion, tutorial, 1d, 1-d, 1D, 1_d, one dimension, 2d, 2-d, 2D, 2-D, two dimension, PDE, Dirichlet, Neumann, Robin, boundary condition, Random Walk, fractional Brownian motion, fBm, fGn, FGP

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.