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.
81
questions
9
votes
1answer
561 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 ...
3
votes
0answers
84 views
Strange Picard iteration
I am interested in solving the equation
$$
\begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
0
votes
1answer
90 views
Free Electron Schrödinger Equation (Energy Method)
For the simplest atom, its wave function is described by the PDE of Schrodinger equation:
$$
-i h \frac{\partial u}{\partial t }=\frac{h^{2}}{2m} \Delta u + \frac{e^{2}}{r}u$$
The potential $\frac{e^{...
3
votes
1answer
146 views
For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?
For the first-order explicit upwind scheme, it can be easily shown that, if one keeps the same grid size and progressively decreases the time step below the max allowed one (i.e. below CFL~1) the ...
0
votes
0answers
42 views
The physical meaning of conservative mass in diffusion equation
I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
3
votes
1answer
714 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 ...
2
votes
0answers
73 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 ...
0
votes
0answers
149 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 ...
1
vote
1answer
84 views
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:...
1
vote
0answers
58 views
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
$$ \frac{\partial{u}}{\partial{t}}(x,t) = \frac{\partial}{\partial{x}}\left(D(u) \...
-1
votes
1answer
211 views
Simulating 1D diffusion
I'm trying to understand the influence of Neumann boundary condition while simulating 1D diffusion equation
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C).
$$
The initial value is set ...
1
vote
0answers
49 views
Unsteady diffusion equation with spatial finite elements and Forward Euler in time
I have solved the unsteady diffusion equation using piece-wise linear Finite elements(triangles) for spatial discretisation and Forward Euler for temporal discretisation. I have the following mesh ...
0
votes
0answers
73 views
Comparison of diffusion time - theoretical value vs computed
This is a follow up to my previous post
I've been trying to compare the diffusion time obtained from theoretical
derivation(answered in my previous post) and what is obtained computationally, for a ...
4
votes
1answer
325 views
Computation of diffusion time
While simulating the diffusion of a substance in 1D,
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C).
$$
I'd like to compute the diffusion time
In this link, the diffusion time is given ...
1
vote
1answer
171 views
Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
5
votes
2answers
227 views
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
I am considering the following diffusion equation:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$
over a grid ...
1
vote
0answers
76 views
Solving diffusion equation using finite difference method
I am solving an 1-dimensional diffusion equation with Neumann boundary condition at outlet and constant concentration, C, at the inlet. In the end, I want to observe how the concentration diffuses ...
1
vote
0answers
36 views
Setting up diffusion with integral B.C. in Fenics
I'm trying to model diffusion through a cylindrical domain $D = \{ (x,y,z) : x^2 + y^2 \leq 1, \;\; 0 \leq z \leq 1\}$.
The is an initial concentration of the diffusant at the upper flat surface, ...
2
votes
1answer
85 views
Numerical methods for non-linear diffusion
I have the following non-linear diffusion equation, for $\ z(x,t)$:
$\ z_t = -C(\sin(\omega t))^m x^{hm}(hm x^{-1}(z_x)^n + n z_{xx} (z_x)^{n-1}) $
Any advice for numerical (or analytical) solutions?...
1
vote
1answer
60 views
What is the right way to set up two random tensor fields which have an identical average diffusivity
I want to compare some properties of traveling waves through two randomly diffusive media. The traveling waves follow the fisher equation:
$$\frac{du}{dt} = \nabla(\mathbf{D}_{\gamma} \nabla u) + u(1-...
0
votes
1answer
269 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 ...
0
votes
1answer
96 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 ...
3
votes
0answers
94 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/...
1
vote
1answer
78 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:
...
0
votes
1answer
127 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 ...
0
votes
1answer
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 ...
2
votes
0answers
113 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 ...
1
vote
1answer
307 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 ...
8
votes
1answer
449 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 ...
0
votes
1answer
31 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 ...
1
vote
0answers
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 ...
0
votes
1answer
49 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 ...
1
vote
0answers
238 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 ...
4
votes
2answers
316 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 ...
3
votes
0answers
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 ...
2
votes
2answers
469 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 ...
2
votes
2answers
709 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)=...
-2
votes
1answer
318 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 ...
3
votes
1answer
146 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 ...
1
vote
0answers
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 ...
1
vote
2answers
174 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 ...
1
vote
1answer
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. ...
1
vote
3answers
493 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 ...
1
vote
0answers
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 ...
3
votes
0answers
216 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}
...
1
vote
0answers
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$$
Of course, this equation can be solved analytically. I am trying to ...
0
votes
1answer
284 views
The system matrix and the right hand side for diffusion equation with staggered grid
In the following staggered grid setting,
I want to solve diffusion equation as a linear system.
$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$...
0
votes
2answers
436 views
Stable implicit method to solve convection-heat diffusion in 3D
The last couple of hours I have been looking for an unconditionally stable method to solve the convection-diffusion equation within a 3D inhomogeneous material.
Here's the well known diffusion-...
1
vote
1answer
98 views
Transforming a 1D cartesian variable-coefficient diffusion code into a 1D adially symmetric one
So I have a code that I use which solves a 1D variable coefficient diffusion problem in cartesian coordinates:
$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(D(x)\frac{\partial u}{\...
1
vote
0answers
237 views
Reaction-diffusion equations
I'm simulating a biological phenomena with reaction diffusion equations. There are multiple diffusing materials and there are some complex relations about consumption and production of such materials. ...
