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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 . $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
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### 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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### 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. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$ ...
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### Interpreting results of using no-flux boundary condition

I am solving for solute transport in 1 D. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$ No-flux boundary condition is imposed at both the ...
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### Inverse advection-diffusion problem, solving for a drift coefficient with experimental data?

I am investigating a physical process where I believe the 1-D advection-diffusion equation: \begin{equation} \frac{\partial u}{\partial t} = -\frac{\partial}{\partial x}[\mu(x,t) u(x,t)] + \frac{\...
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### Solving PDE with spatial and temporal derivatives on left hand side

I wish to solve an equation of the form, $$\frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F})$$ for the variable $\phi$ (e.g. ...
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### 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 ...
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### 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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### 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 ...
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### How to formulate Poisson's equation into flux eqution

I have a small 2D system I'm trying to model using a non-linear extension of Darcy's law for fluid flow in porous media. I'm primarily interested in the local flow velocity, not necessarily the ...
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### Can this nonlinear advection-diffusion equation be discretized as to only have to solve SPD systems?

Consider a nonlinear advection-diffusion equation of the form $$\frac{\partial u}{\partial t} = \nabla \cdot (a(u) \nabla b(u) - \vec{c}(u)u) \tag{1}$$ on a rectangular domain with Dirichlet ...
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### Solve ODE with two unknown functions

I want to solve a diffusion-convection problem numerically. I start from a PDE of the form $$\partial_t f(x,t) = \partial_x(K(x)\;f(x,t))+\partial_{xx}\;f(x,t)^{m}\;.$$ It is possible to calculate a ...
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### When is it safe to ignore the diffusion term in an advection-diffusion equation?

Given the one dimensional equation: $\epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0$ with $0\le\epsilon \ll1$ with boundary conditions $u(0) = 0$ and $u(1) = 2$, we ...
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### How can I numericaly solve a convection-diffusion equation with a large diffusion term?

I want to numerically solve the advection-diffusion equation: \begin{equation} u_t(x,t) + cu_x(x,t) = v u_{xx}(x,t) \end{equation} for $x \in [0,1]$ and $t \geq 0$ subject to the boundary conditions ...
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