# 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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### Numerically solving the Advection-diffusion equation with no-flux boundary condition leads to violation of mass conservation

I am trying to solve numerically the advection-diffusion equation of the following form $$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$ ...
• 11
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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,...
140 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 ...
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### 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, ...
1 vote
207 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 ...
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1 vote
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### 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 ...
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• 61
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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 ...
• 61
110 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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1 vote
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 ...