Questions tagged [advection-diffusion]

Questions related to solving the advection-diffusion equation using numerical methods, including derivation and implementation of boundary conditions.

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How to quantify the numerical diffusion term in a second-order upwind advection scheme?

In the first-order upwind scheme, numerical diffusion can be quantified as: $$\frac{dT}{dt} = -w\left(\frac{dT}{dz}\right) + \left(\frac{wdz}{2}\right)\left(\frac{d^2T}{dz^2}\right)$$ For Lax-Wendroff,...
Yoni Verhaegen's user avatar
5 votes
1 answer
126 views

Slope limiting with implicit time integration

I am solving the advection problem with high order numerical methods, using the method of lines. The boundary conditions and initial condition are selected in a way where I know that the exact ...
vainia's user avatar
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Implementation of operator splitting method for Wigner equation

I am dealing with the integro-differential equation for Wigner function, $$\frac{\partial f}{\partial t}+p\frac{\partial f}{\partial x}+\\+\frac{1}{\chi}\left\{\int_{-\pi}^{+\pi}dy\,\int_{-\infty}^{+\...
Artem Alexandrov's user avatar
1 vote
2 answers
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Modeling contamination diffusion in a draining container, part 2

Part 1, but I'll repeat here. This time we'll move the top boundary. I have a container of water with initial volume $V_0$ and height $L$, open at the top which receives a constant mass flux $\phi$. A ...
HiddenBabel's user avatar
2 votes
1 answer
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Modeling contamination diffusion in a draining container

I asked this on the Physics site, but it fits much better here, and I now can ask a more specific question. For the scope of the problem, I have a 250-mL bottle filled with pure water and a sampling ...
HiddenBabel's user avatar
3 votes
1 answer
206 views

First derivative central differences with reflecting boundary conditions

I have the following problem in 1-D: \begin{equation} \partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega. \end{...
lightxbulb's user avatar
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3 votes
2 answers
239 views

Solving systems of advection-diffusion-reaction equations with finite element methods

I have been doing a lot of self-study on numerically solving PDEs so that I can solve system of linear and nonlinear Advection-Diffusion-Reaction (ADR) systems on complex meshes. I have been watching ...
krishnab's user avatar
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3 votes
2 answers
197 views

Non-conservative advective term in a finite volume scheme

I am interested in solving this set of nonlinear couples advection-diffusion equations using a finite volume scheme: $$ \frac{\partial f(x,y)}{\partial t}=-(\boldsymbol{u}+\nabla\eta)\cdot\nabla f +\...
BitterDecoction's user avatar
0 votes
1 answer
140 views

Choice of grid generation for FDM discretisation methods

I'm currently revisiting some FDM schemes for convection-diffusion equations in 1D, 2D and 3D and getting up to speed with the industry-standard methods again. The application is derivatives pricing, ...
freistil90's user avatar
1 vote
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182 views

Solving PDE on a non-uniform grid with Crank-Nicolson scheme

I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
ottavio 's user avatar
1 vote
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232 views

Closed (Robin) boundaries in advection-diffusion equation with FDM

I am solving the equation $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{\partial \phi}{\partial x} + v\phi \right) $$ using finite differences. I want to include ...
rugermini's user avatar
2 votes
1 answer
144 views

Solving a set of mixed conservative/non-conservative equations with the finite volume method

I want to solve this set of 2D advection-diffusion equations of this form in spherical coordinates: $$ \frac{\partial f}{\partial t}=-\mathbf{u}\cdot\nabla f+\eta\nabla^2f+\eta_1(\mathbf{e}_1\cdot\...
BitterDecoction's user avatar
2 votes
1 answer
715 views

Why is my Runge-Kutta 4 solution to the 1-D advection equation decaying so quickly?

I am trying to numerically solve the advection equation $y_t + y_x = 0$ using a the "classical" Runge-Kutta 4 explicit timestepping method, along with a left-hand finite difference ...
bosco98's user avatar
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3 votes
1 answer
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Mineral dissolution and solute transport around a solid

I am trying to simulate solute transport of acid (HCl) and consequent mineral dissolution around a grain (calcite). The governing equation for transport is the advection-diffusion equation, given as: ...
oma11's user avatar
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0 answers
134 views

Advection diffusion equation using Crank-Nicolson with total flux and Diriclet BCs

I am trying to model the 1D advection-diffusion equation: $${\partial c \over \partial t} = D_c{\partial^2 c \over \partial x^2} -u{\partial c \over \partial x}.$$ With Robin boundary conditions that ...
DozerD's user avatar
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Solute transport around a solid obstacle

I am a newbie in CFD and single/multiphase flow and transport in general. As part of my quest to learn, I am trying to model solute transport around a solid object in the center of a 2D domain. The ...
oma11's user avatar
  • 55
3 votes
1 answer
86 views

Instability at the boundary of a finite difference simulation of a hyperbolic PDE

I want to simulate the hyperbolic partial differential equation $$W_{tt} + V W_{tx} + k_E V W_x + k W_t = 0,$$ but I am having trouble finding a discrete analog of this equation which is numerically ...
kevinkayaks's user avatar
0 votes
1 answer
112 views

Manufactured solution to 2d convection-diffusion with homogeneous Robin boundary conditions

I am looking for a manufactured (or analytical if it exists) solution to the 2d boundary-value problem $$\frac{\partial u}{\partial t} = \mathbf{a} \cdot \nabla u + D \nabla^2 u \quad \quad \mbox{in } ...
Max_89's user avatar
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2d advection-diffusion: cell Péclet number and numerical stability

I am studying the numerical resolution of 2d advection-diffusion problems with finite element methods. $$\frac{\partial u}{\partial t} + \beta\cdot\nabla u = \nabla\cdot(\nabla u) \, .$$ It is said in ...
Max_89's user avatar
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2 votes
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108 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 ...
Natasha's user avatar
  • 421
2 votes
2 answers
740 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 ...
Natasha's user avatar
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0 answers
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Why is this Advection-convection model with insulating boundary losing mass?

I am trying to model a 1-d advection-convection numerically, using an upwind scheme. I'm using the following equation to calculate the value of internal cells: $$C_x^{t+1} = C_x^{t} + D\frac{\Delta t}{...
JMenezes's user avatar
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333 views

Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations

I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
BlaB's user avatar
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2 votes
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33 views

Semi-analytical/empirical modelling of wall boundary conditions in advection-diffusion-reaction equation with distributed source

Let's suppose I need to numerically solve a 3D steady-state transport equation of the form $$ \nabla \cdot (\mathbf{u} c) = \nabla \cdot (D \nabla c) - \lambda c + S $$ where $c$ is the transported ...
adironco's user avatar
2 votes
0 answers
85 views

Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion

I am trying to solve the following coupled partial differential equations with a finite difference scheme: $$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$ $$\partial_tW+v\partial_zW-\...
Hanno Jacobs's user avatar
0 votes
1 answer
393 views

Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization

i am implementing a Matlab code to solve the following equation numerically : $$ (\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z}) $$ with ...
Ivan's user avatar
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-1 votes
1 answer
224 views

Numerical solution of the advection equation with Crank–Nicolson finite difference method

I need to implement a numerical scheme for the solution of the one-dimensional advection equation $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$ \\ C(x,t) = \...
Winabryr's user avatar
2 votes
1 answer
247 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$...
Vefhug's user avatar
  • 309
2 votes
1 answer
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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) = ...
andereBen's user avatar
  • 153
1 vote
0 answers
52 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
cos_theta's user avatar
  • 451
4 votes
1 answer
4k 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 ...
Babaji's user avatar
  • 195
1 vote
0 answers
49 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 ...
Natasha's user avatar
  • 421
0 votes
1 answer
205 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 . $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
Natasha's user avatar
  • 421
1 vote
1 answer
300 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 ...
mfnx's user avatar
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1 vote
1 answer
200 views

Simulating advection - diffusion problem in a network of 1D pipe

I'm interested in solving the following advection-diffusion system in a 1D network of pipes. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$ ...
Natasha's user avatar
  • 421
3 votes
1 answer
1k views

Implementing Robin Boundary condition (finite difference)

I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D. In the following system, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
Natasha's user avatar
  • 421
1 vote
1 answer
2k views

Imposing periodic boundary condition for linear advection equation - Node problem

I've spent the whole day trying to figure out what is the correct way to impose (and implement) periodic boundary conditions $u(0,t)=u(1,t)$ for all $t>0$ for the simple advection equation $u_t + v ...
FEGirl's user avatar
  • 333
2 votes
1 answer
619 views

Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation

I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this $$ \frac{\partial U}{\partial t} + ...
Tan Phan's user avatar
2 votes
1 answer
430 views

1-D boundary value problem: How implement mixed boundary conditions using a FD method?

I have been given a convection-diffusion ODE modeling the steady state temperature of a pipe (through which flows a fluid) as $$-\frac{d}{dz}\left(\kappa \frac{dT}{dz} \right)+v\rho C\frac{dT}{dz}=Q(...
SimpleProgrammer 's user avatar
1 vote
0 answers
38 views

Finite difference/element method : time step and spatial resolution close to a finite singularity

I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same. Let's assume we have this equation : $$\partial_t c - u\...
J.A's user avatar
  • 171
1 vote
0 answers
180 views

FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
Migalobe's user avatar
  • 111
1 vote
1 answer
246 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 ...
Smilia's user avatar
  • 468
1 vote
3 answers
1k views

How to make a less diffusive code to solve 2D advection equation?

I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...
Peanutlex's user avatar
  • 219
2 votes
1 answer
783 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 ...
Natasha's user avatar
  • 421
0 votes
1 answer
157 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 ...
Userhanu's user avatar
  • 151
0 votes
1 answer
2k views

Analytical solution of 1D advection -diffusion equation

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with Neumann boundary condition at both the inlet and outlet of a cylinder through which the fluid flow occurs. $$...
Natasha's user avatar
  • 421
3 votes
1 answer
2k views

Impose Neumann Boundary Condition in advection-diffusion equation 1D

when solving the advection equation in 1D that is: $$ \frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0 $$ with $ u'(t,0) = 0$ and $u(t,L) = 0$ , $u(0,x) = u_{0} $ one numerical ...
tnt235711's user avatar
  • 273
1 vote
0 answers
545 views

Inflow and outflow boundary conditions for advection-diffusion equation

I'm trying to solve this advection-diffusion equation (ADE): $$\frac{\partial \phi}{\partial t} + \nabla \cdot (-D \nabla \phi + \mathbf{u} \phi) = 0$$ In fact, this ADE framework is coupled to a ...
GGG's user avatar
  • 173
1 vote
1 answer
694 views

How write a integration loop in fortran, leapfrog scheme to solvind PDE (advection)?

I want to resolve numerically this equation using of difference finite method with Leapfrog Scheme $$\frac{\partial{u}}{\partial t}+ v \frac{\partial{u}}{\partial x}= 0 $$ I'm trying to write a code ...
PCat27's user avatar
  • 11
5 votes
1 answer
748 views

Finite Differencing schemes for Convection-Diffusion equation

I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger. The flow/convection is always 1D,...
JE_Muc's user avatar
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