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

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### 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. ...
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### 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-...
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### 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-...
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### Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?

I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
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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: \frac{\partial u}{\partial t} = -\frac{\partial}{\partial x}[\mu(x,t) u(x,t)] + \frac{\...
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### 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?" ...
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### 9-point stencil finite difference Laplacian with variable diffusion coefficients

So I'm trying to implement a 9-point stencil discretization to the 2D difussion equation. The stencil is here. However, most of the literature deals with a Laplacian that has a constant diffusion ...
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### 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 ...
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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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### In what regime do the continuous and discontinuous Galerkin method become unstable for advection-diffusion systems?

I know that the finite volume method (based around a central different stencil) is unstable for advection dominated advection-diffusion problems. This leads to different adaptive schemes to can be ...
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### 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=...
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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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I am implementing a finite difference method in solving the diffusive-advective equation: $$u_t + v \cdot u_x = D\cdot u_{xx}$$ (v, D are constants). Planning to use the operator splitting method (...