Questions tagged [advection-diffusion]
Questions related to solving the advection-diffusion equation using numerical methods, including derivation and implementation of boundary conditions.
18
questions
17
votes
1
answer
1k
views
How do you debug numerical code, what could be source of this oscillatory error?
Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
- 5,359
2
votes
1
answer
233
views
Finite element (1D) for steady state non-linear problem
I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0$...
- 289
5
votes
2
answers
3k
views
Does the finite element method impose any restrictions on the Peclet number for numerical stability?
Background on finite volume method
When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
- 5,359
20
votes
1
answer
4k
views
BDF vs implicit Runge Kutta time stepping
Are there any reasons for why one should choose high order implicit Runge Kutta (IMRK) over BDF time stepping? BDF seems much easier to me as $q$ stage IMRK needs $q$ linear solves per time step. ...
- 587
13
votes
3
answers
1k
views
What are the basic principles behind generating a moving mesh?
I am interested in implementing an moving mesh for an advection-diffusion problem. Adaptive Moving Mesh Methods gives a good example of how to do this for Burger's equation in 1D using finite-...
- 5,359
8
votes
1
answer
1k
views
Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?
Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as,
$$
w_j^{\prime} = \frac{w_{...
- 5,359
6
votes
2
answers
2k
views
Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?
I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?"
...
- 5,359
4
votes
1
answer
3k
views
Finite difference methods in cylindrical and spherical co-ordinate systems
I am quite familiar with finite difference schemes in cartesian coordinates. The key point here is that every point in the cartesian grid is treated equally as the spacing between consecutive points ...
- 195
3
votes
3
answers
659
views
Numerical approximation for a known exact solution of advection-dispersion equation
My goal is to create a numerical solution of 1D-solute transport (Convective-dispersion equation, CDE) to match it's analytical solution based on experimental data. The CDE can be written as (where, C=...
- 143
3
votes
2
answers
2k
views
Does the time-dependent advection-diffusion equation have an analytical solution?
The advection-diffusion problem, where $0<x<1$,
$$u_t = (-au + du_x)_x$$
with Dirichlet boundary conditions,
$
u(0)=1,~u(1)=0
$
, has the steady-state solution,
$$
u(x) = \frac{e^{\lambda} - ...
- 5,359
2
votes
1
answer
671
views
Analytical Solution of Transport Equation
I'm looking at the analytical solution of the convection-diffusion equation
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$
with initial ...
- 461
2
votes
1
answer
746
views
Weak formulation for advection diffusion reaction
I need a check on the following exercise about weak formulations and finite elements.
Consider the advection diffusion system
$$
\begin{cases}
-(\mu u')' + \beta u' + \gamma u = f \\
u(a)=0 \\
u(b) = ...
- 153
2
votes
0
answers
84
views
Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?
I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
- 461
2
votes
2
answers
579
views
How to implement point source or volume source in finite element implementations
I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe.
I have points positioned along the length of the pipe (blue dots in the image above).
I ...
- 461
1
vote
0
answers
48
views
Comparison of convection time - theoretical value vs computed
This is a follow up to my previous post here,
I'm solving for convection in 1D
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
The discretization of the above equation is ...
- 461
1
vote
1
answer
253
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 ...
- 172
1
vote
1
answer
170
views
Error for the finite differences scheme -- Advection equation
Consider the advection equation (1D in space)
$$
\frac{\partial u}{\partial t} + V\, \frac{\partial u}{\partial x}=0
$$
and we solve it numerically on $[0,1]\times [0,1]\ni (t,x)$ using a forward ...
- 269
0
votes
1
answer
189
views
Question on comparing the accuracy of numerical schemes
This is a follow up to my previous post here
I'm solving the following 1D transport equation .
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
- 461