Questions tagged [diffusion]

For questions about modeling movement from high to low concentrations, where concentration could, for example, refer to number of particles or amount of thermal energy in a region. Problems will often involve solving a diffusion (differential) equation.

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Calculating the species mass consumption from implicit reaction-term in diffusion-reaction equation

The 1D diffusion equation with a chemical source term has the following form: $$\frac{\partial Y}{\partial t} = D \frac{\partial^2 Y}{\partial x^2} - k Y,$$ where $Y$ is the molar concentration of the ...
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2 votes
3 answers
79 views

Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

I'm dealing with the numerical resolution of advection-diffusion-reaction equations. I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations. I'm ...
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Finite volume method for 1D heat equation in 1D

I wish to solve the following using the finite volume method: $$\frac{\partial u}{\partial t}=\frac{D}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+Q(t,r)$$ with the ...
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9 votes
1 answer
650 views

How to solve a second order differential equation (diffusion) with boundary conditions using Python

I am having trouble implementing a model from a publication. Huang, K-L.; Holsen, T.M.; Selman, J.R. Ind. Eng. Chem. Res. 2003, 42, 15, 3620–3625 scihub link: https://sci-hub.se/10.1021/ie030109q I ...
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3 votes
0 answers
91 views

Strange Picard iteration

I am interested in solving the equation $$ \begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
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1 answer
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Free Electron Schrödinger Equation (Energy Method)

For the simplest atom, its wave function is described by the PDE of Schrodinger equation: $$ -i h \frac{\partial u}{\partial t }=\frac{h^{2}}{2m} \Delta u + \frac{e^{2}}{r}u$$ The potential $\frac{e^{...
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3 votes
1 answer
196 views

For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?

For the first-order explicit upwind scheme, it can be easily shown that, if one keeps the same grid size and progressively decreases the time step below the max allowed one (i.e. below CFL~1) the ...
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The physical meaning of conservative mass in diffusion equation

I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads $$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
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3 votes
1 answer
866 views

Solving the heat diffusion equation with source term

I am trying to solve the 1-D heat equation numerically with a variable source term. The system is basically a tank containing styrene in which it polymerizes to liberate heat. I have assumed that the ...
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2 votes
0 answers
92 views

Modelling of Stefan Maxwell equation

I am trying to solve Maxwell Stefan's equation over a membrane to get the transient mole fraction distribution over the membrane thickness 'z'. But somehow I am not able to code it using ODE45, more ...
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254 views

2D diffusion equation using Finite Volume Method

i am working on an assignment problem: Consider a two-dimensional rectangular plate of dimension L = 1 m in the x direction and H = 2 m in the y direction. The plate material has constant thermal ...
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1 answer
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FEM diffusion: inaccurate results small time steps

I wrote some FEM code and found some strange results when using a very small time step. So, I decided to analyze the discrete equations. Consider the following linear diffusion problem in 1 dimension:...
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How to achieve (approx) unit scaling of a non-linear diffusion (heat) equation with a wildly varying diffusion coefficient?

I have numerical issues with a poorly scaled one-dimensional non-linear diffusion equation in physical co-ordinates $$ \frac{\partial{u}}{\partial{t}}(x,t) = \frac{\partial}{\partial{x}}\left(D(u) \...
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-1 votes
1 answer
232 views

Simulating 1D diffusion

I'm trying to understand the influence of Neumann boundary condition while simulating 1D diffusion equation $$ \frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C). $$ The initial value is set ...
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Unsteady diffusion equation with spatial finite elements and Forward Euler in time

I have solved the unsteady diffusion equation using piece-wise linear Finite elements(triangles) for spatial discretisation and Forward Euler for temporal discretisation. I have the following mesh ...
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74 views

Comparison of diffusion time - theoretical value vs computed

This is a follow up to my previous post I've been trying to compare the diffusion time obtained from theoretical derivation(answered in my previous post) and what is obtained computationally, for a ...
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4 votes
1 answer
330 views

Computation of diffusion time

While simulating the diffusion of a substance in 1D, $$ \frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C). $$ I'd like to compute the diffusion time In this link, the diffusion time is given ...
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1 vote
1 answer
186 views

Crank-Nicholson for diffusion-advection vs diffusion equation

Let's consider the following 1D diffusion equation: $\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$ where we assume that the diffusion ...
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5 votes
2 answers
248 views

Is the diffusion equation with Neumann and Dirichlet BCs well-posed?

I am considering the following diffusion equation: $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$ over a grid ...
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0 answers
81 views

Solving diffusion equation using finite difference method

I am solving an 1-dimensional diffusion equation with Neumann boundary condition at outlet and constant concentration, C, at the inlet. In the end, I want to observe how the concentration diffuses ...
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  • 511
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0 answers
39 views

Setting up diffusion with integral B.C. in Fenics

I'm trying to model diffusion through a cylindrical domain $D = \{ (x,y,z) : x^2 + y^2 \leq 1, \;\; 0 \leq z \leq 1\}$. The is an initial concentration of the diffusant at the upper flat surface, ...
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  • 111
2 votes
1 answer
89 views

Numerical methods for non-linear diffusion

I have the following non-linear diffusion equation, for $\ z(x,t)$: $\ z_t = -C(\sin(\omega t))^m x^{hm}(hm x^{-1}(z_x)^n + n z_{xx} (z_x)^{n-1}) $ Any advice for numerical (or analytical) solutions?...
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1 answer
60 views

What is the right way to set up two random tensor fields which have an identical average diffusivity

I want to compare some properties of traveling waves through two randomly diffusive media. The traveling waves follow the fisher equation: $$\frac{du}{dt} = \nabla(\mathbf{D}_{\gamma} \nabla u) + u(1-...
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0 votes
1 answer
293 views

Numerically solving a non-linear PDE

I have this non-linear partial differential equation. $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ I want to use the finite ...
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0 votes
1 answer
99 views

Stability of PDEs

I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf). The largest stable timestep that can be taken for this ...
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3 votes
0 answers
100 views

Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
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1 vote
1 answer
81 views

Minimizing the used memory in diffusion simulation using Python

I am recently dealing with a diffusion simulation project and I have come up with the following code: ...
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0 votes
1 answer
132 views

Simulating Brownian motion in 3-D for first hitting time?

I want to simulate Brownian motion in 3-D for the following conditions: $$p(x=0,y=0,z=0,t=0)=1$$ $$p(x,y,z=c,t)=0$$ where $p$ is the probability of finding molecules in the 3-D environment. I want to ...
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0 votes
1 answer
4k views

Applying neumann boundary conditions to diffusion equation solution in python [duplicate]

For the diffusion equation $$ \frac{\partial u(x,t)}{\partial t} = D \frac{\partial ^2 u(x,t)}{\partial x^2} + Cu(x,t) $$ with the boundary conditions $u(-\frac{L}{2},t)=u(\frac{L}{2},t)=0$ I've ...
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2 votes
0 answers
118 views

Harmonic average of Diffusion Tensors in Finite Volume Method

I want to implement a Bilinear Finite Volume discretisation of the anisotropic diffusion problem: $$\frac{du}{dt} = \nabla \cdot (\textbf{D} \nabla u)$$ Both my degrees of freedom as well as the ...
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1 vote
1 answer
329 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D. In order to both test the timestepping and the spatial discretisations I had a look at using ...
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8 votes
1 answer
464 views

Computing geodesic distances with diffusion

I am trying to solve an APSP (All-Pair Shortest Path) problem on a weighted graph. This graph is actually a 1, 2 or 3 dimensional grid, and the weights on each edge represent the distance between its ...
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0 votes
1 answer
32 views

Calculating volume of a discretised diffuse interface object

Suppose I have a spherical object projected onto a discrete square mesh. The dicretised circle can be represented by filling a logical matrix such that voxels in the interior of the sphere are filled ...
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1 vote
0 answers
45 views

Limit to volume change in a discretized mathematical model?

I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through ...
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0 votes
1 answer
51 views

Solving the diffusion/heat equation for a randomly distributed set of points in 3D

In this problem I am trying to solve, I have a messy set of points distributed in 3D space, each with a defined temperature. If I would want to calculate the heat transfer scenario in this system, how ...
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1 vote
0 answers
244 views

Neumann boundary condition FD implementation for instationnary diffusion equation

I am trying to solve this diffusion equation : $\dfrac{\partial D\dfrac{\partial f}{\partial x}}{\partial x}+S = \dfrac{\partial f}{\partial t}$ ($D$ is not constant and varies according to $x$) with ...
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4 votes
2 answers
335 views

Mean-squared displacement in Monte Carlo studies

Is measuring mean-squared-displacement in Monte Carlo simulations uncommon? I'm very interested to find out if this has actually ever been tried. For instance, in the context of spheres, or ...
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3 votes
0 answers
109 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
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2 votes
2 answers
509 views

Conservation violation in axisymmetric Diffusion Equation

1d diffusion equation Integrating the diffusion equation, $$ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}, $$ with a constant diffusion coefficient D using forward Euler for ...
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  • 131
2 votes
2 answers
761 views

Neumann boundary conditions diffusion equations methods of lines

I want to solve the diffusion equation using the method of lines with Neumann boundary conditions $$ \frac{\partial p}{\partial t}=\frac{\partial^2p}{\partial x^2}\\ \frac{\partial p}{\partial x}(x=0)=...
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-2 votes
1 answer
327 views

Adding Non-Linear source term to 2d Implicit MATLAB code

I'm running out of time for this code so any help would be greatly appreciated. I am currently coding the 2D heat/diffusion equation in matlab but i'm having trouble adding in the source term. my ...
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3 votes
1 answer
151 views

Diffusion properties of hard spheres in Monte Carlo simulation

In standard Monte Carlo simulations, say for hard sphere systems, how should one compute the mean-squared displacement of the spheres in order to extract dynamical properties such as the diffusion ...
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1 vote
0 answers
132 views

Hello word in FEniCS? [closed]

I am trying to start using FEniCS, but have a problem with the simple hello world examples given in the books. Could you please give me the simplest hello world ...
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1 vote
2 answers
177 views

Computational Physics: Finding the Diffusion Coefficient from the Discretized Diffusion Equation

I'm pretty new to translating simulation to reality so please forgive the perhaps naive approach I'm taking here. If we have a (quasi-2D) experimental video of a certain concentration changing with ...
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1 vote
1 answer
6k views

How to simulate 3D diffusion in python?

I want to simulate a simple 3D diffusion (e.g., an ink released from one side of a vessel) using SciPy. There are some tutorials for one-dimensional diffusion. ...
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1 vote
3 answers
508 views

2D simulation of a particle with different diffusion coefficient in different directions of the particle

How can one simulate diffusion of a particle which has a natural axis and diffusion coefficient of the particle in the direction of the axis is D_1 and in perpendicular to the axis is D_2? Could ...
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1 vote
0 answers
55 views

How is the Gastner-Newman equation implemented to create value-by-area cartograms?

There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
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3 votes
0 answers
226 views

Spurious oscillations in diffusion-reaction problems with finite volume

I have successfully solved the multi-species diffusion-reaction equation \begin{equation} \frac{\partial c_i}{\partial t} = \nabla \cdot (d_i(x)\nabla c_i) + s_i(x,t), \quad \quad (1) \end{equation} ...
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1 vote
0 answers
58 views

elliptic equation with exponential coefficient

I'm trying to solve the following equation $$\dfrac{\partial}{\partial x}\left(e^{au}\dfrac{\partial u}{\partial x}\right) = 0$$ Of course, this equation can be solved analytically. I am trying to ...
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0 votes
1 answer
290 views

The system matrix and the right hand side for diffusion equation with staggered grid

In the following staggered grid setting, I want to solve diffusion equation as a linear system. $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$...
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