Questions tagged [advection]
To move in some direction (as a fluid does in a pipe). Often contrasted with diffusion, which is a spreading out without necessarily having any movement of the field as a whole.
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Crank-Nicolson scheme in space for advection equation
Consider the equation
$$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$,
for $t,x\in\mathbb{R}$.
I'd like to solve this equation forward in space and backward in time, ...
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Neumann boundary conditions for the upwind scheme applied to the advection equation (Python)
I'm trying to solve the linear advection equation
$$u_{t} = cu_{x}, \\
x \in [x_{0}, x_{e}], \quad t \in (0, T], \quad c \in \mathbb{R} \\
u(x,0) = f(x)$$
Note that for $c > 0$, the solution is a ...
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Why can I not solve the negative advection equation (backwards in time)?
Suppose we have the negative, inhomogeneous advection equation:
$$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
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Crank-Nicolson method for inhomogeneous advection equation
Suppose we have the inhomogeneous advection equation
$$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$
for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
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Discontinuos Galerkin Method - inhomogeneous advection problem
I'm currently trying to get into this topic. I've learned that the basic scheme for the advection problem ($D_{x}u+a*D_{t}u=0$) can be solved in a scheme like $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}...
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Deposition model in laminar flow
I have a chamber full with a fluid flowing horizontally in laminar regime from one side to the other. It carries a suspension with concentration $c$. This suspension also falls to the bottom of the ...
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Numerical diffusion in during advection of a free surface in an FE context
I am currently working on a project where a two-phase flow is considered. The phases are described using a level set approach and a signed distance function from the interface between the phases where ...
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When is it safe to ignore the diffusion term in an advection-diffusion equation?
Given the one dimensional equation:
$\epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0 $
with $0\le\epsilon \ll1$ with boundary conditions $u(0) = 0$ and $u(1) = 2$, we ...
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Simple steady-state advection problem: do I need FVM with upwind scheme?
I have a 2D (x,y) scalar advection problem that describes net blowing snow ($q$) transport at a point. This takes the form
$$q = A - F*\nabla\cdot(q {\bf \hat u}),$$
where A ($kg\cdot m^{-2}\cdot s^{...
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Upwind difference for velocity in staggered grid
I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf
In the paper, the nonlinear term is treated as mix of central central difference and upwind difference using a ...
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How to support or contradict a hypothesis on unconditional stability using numerical optimization
The main motivation behind my next question is that I think I derived a higher order numerical scheme for linear advection equation that is unconditionally stable using Von Neumann stability analysis.
...
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Spherical Advection Discretization (boundary nodes)
Consider the spherical advection problem: describing the conservation of a property $u$ in a closed spherical domain.
$$
\frac{\partial u}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^...
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Direction-splitting for SSP-RK schemes
What are the implications of applying a direction-splitting within each stage of an SSP-RK scheme?
For instance, given a standard advective transport type equation:
$$
\partial_{t}Q + \operatorname{...
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More Smearing with decreasing timestep in advection problems
I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate).
Let there be a equally ...
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Corner Transport Upwind for Linear Advection in Arbitrary Velocity Field
I need to implement a 3D version of the Corner Transport Upwind (CTU) finite volume method (in python); and so I've been reading Leveque, "Finite Volume Methods for Hyperbolic Problems" which I think ...
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Numerical solution of non-linear advection equation other than inviscid burgers
I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
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Numerically computing the advection equation
I am trying to write a program to compute the advection equation.
$$u_t +u_x = 0$$
I use the spectral method for the spatial derivative $u_x$ and the leapfrog method for the time derivative $u_t$.
...
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Discretization method for advection equation without numerical diffusion
Given the advection equation for an incompressible flow field
$$\frac{\partial c}{\partial t} + \mathrm{Pe} \frac{\partial c}{\partial x} = 0$$
what would the best method be for discretizing this ...
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boundary conditions of linear advection problem
I am solving the 1D advection problem given by:
$$\frac{\partial u}{\partial t}=-c\frac{\partial u}{\partial x}$$
where c is the wave speed, and u is the unknown field variable, and x and t are time ...
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Find cfl condition
We have the advection equation $u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R}$ with initial condition $u(0,x)=u_0(x)$.
Suppose that we have the following sheme:
I want to find the CFL ...
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Advection equation using the finite element method
I want to solve this simple advection equation using the finite element method.
$$\frac{dc}{dt}+v\cdot\nabla c = f$$
What's the best FEM discretization for this? I have tried using the standard/...
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what do zero real parts of eigenvalues mean? Any good references?
I am solving a 1D advection problem of the the form
$$dQ/dt=[A]Q$$
where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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what do positive real parts of eigenvalues mean?
I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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Calculating theoretical order of accuracy of least squares fit advection scheme
I'm familiar with finding the order of accuracy using von Neumann analysis for finite difference schemes formulated using Taylor series expansions.
But is there a similar technique for finding the ...
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Implementation of 1D Advection in Python using WENO and ENO schemes [closed]
I'm trying to implement 1D advection solver using WENO and ENO schemes.
\begin{equation}
\frac{\partial u}{\partial t}
+ \frac{\partial f(u)}{\partial x} =0
\end{equation}
where:
\begin{...
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CFD: Doubt with time convergence in advection fully implicit upwind scheme
I'm trying to solve an advection - convection problem using an implicit upwind scheme - you can see here the finite difference discretization used.
I start the model (built from scratch on Scilab) ...
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Usability of upwind finite difference schemes
NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here.
Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the ...
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How to calculate numerical dispersion relations for Spectral Elements?
How can we determine the numerical dispersion relation of a Spectral Element Method which leads to coupled systems of algebraic equations?
What approaches to do analysis of dispersion relations is ...
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Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?
Suppose I have a 1D advection equation in conservation (divergence) form
$\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
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How can I prove numerical diffusion in upwind scheme for transport equation
I was just implementing the upwind scheme for a linear transport equation $u_t + cu_x = 0$ where $c=0.5$ and I saw that the solution was indeed advected but over time it starts to diffuse. Can anyone ...
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Finite differences scheme for 2D advection equation
I'm trying to study with a finite difference method the 2D advection equation with a space-dependant flow.
Taking a function $f(x,y,t)$ solution of the equation :
$$
\partial_t\,f+\nabla(\textbf{v}\,...
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Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions
I would like to numerically solve a hyperbolic PDE of the form
$\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
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advection diffusion equation
Would you know what is the condition for stability for the advection-diffusion equation where we treat the diffusion part using Crank-Nicholson and the advection part using FCTS (forward in time ...
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Implementation of convection scheme given by normalized variable diagram
In finite difference and finite volume methods, convection schemes (upwind, central, quick, ...) are usually shown in a normalized variable diagram. The diagram gives the normalized face variable as a ...
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Finite Differencing of a Strange Advection-Reaction Problem
comp! I'm trying to solve the advection-reaction problem
$ dg/dt = dg/dx + x\cdot g \qquad on~~x \in \Omega = (-\infty, +\infty)$
supplemented with the boundary conditions
$ \lim_{\lvert x \rvert \...
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Simulating advection over a network of 1D pipes
I am trying to create a simulation to help visualize how different chemical components flow through a network of pipes with associated valves, pumps, and chemical inputs. In this simulation, the pipes ...
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How to discretize the advection equation using the Crank-Nicolson method?
The advection equation needs to be discretized in order to be used for the Crank-Nicolson method. Can someone show me how to do that?
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Conservative finite-difference expression for the advection equation
Following on from the earlier question I am trying to derive a finite-difference scheme for the advection equation which is conservative. It was suggested that for advection equation with variable ...
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Is stabilization of energy equation needed when momentum equation needs it?
When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets ...
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Can the advection equation with variable velocity be conservative?
I am trying to understand the advection equation with variable velocity coefficient a bit better. In particular I don't understand how the equation can be conservative.
The advection equation,
$$
\...
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Stabilization of convection-dominated flow and turbulence modeling
Are stabilization techniques for convection-dominated flows like SUPG+PSPG, interior penalty methods, etc. able to handle turbulent flows without tubulence model being employed, at least up to some ...
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Are the drift-diffusion equations from semiconductor physics analogous to solving an advection-diffusion problem?
I am trying to understand an extra terms that appears when I derive the drift-diffusion equations for semiconductors. The extra term (see below) comes from applying the chain rule to the advection ...
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Open boundary conditions with the advection-diffusion equation
Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term),
$$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\...
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Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation
I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
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Strange oscillation when solving the advection equation by finite-difference with fully closed Neumann boundary conditions (reflection at boundaries)
I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be ...
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Is Crank-Nicolson a stable discretization scheme for Reaction-Diffusion-Advection (convection) equation?
I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection ...
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code for surface advection (e.g. level set advection)
I have a 2D surface in 3D that I want to advect under a velocity field. More precisely, I have a surface $S$ and a velocity field $v$ and I want to advect $S$ under $v$ using the flow map of $v$, i.e....
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How to solve advection equation using semi-lagrangian method?
I am working on something that involves solving an advection equation $\partial{x}/\partial{t}+\vec{u}\cdot\nabla{x}=0$ in 3D. I discretized the space into 3d cartesian grid and used the Semi-...
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Boundary conditions for the advection equation discretized by a finite difference method
I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs.
The books and notes which I currently have access to all say ...
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How can I derive a bound on the spurious oscillations in the numerical solution of the 1D advection equation?
Suppose I had the following periodic 1D advection problem:
$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a ...
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