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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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Efects from the boundary in advection equation [duplicate]

I am implementing the advection equation $u_x+(1/c)u_t=0$ following a Crank-Nicholson finite difference scheme. The equation for this is \begin{eqnarray*} -\frac{\gamma}{4} w_{n-3 j+1} + w_{n-2 j+1} ...
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Trapezoidal Method applied to the advection equation: find $A$ in $U'(t) = AU(t)$

I have the following method of lines discretization of the advection equation: $$ U_j^{n+1}=U_j^n - \frac{ak}{2h}(U_j^n - U_{j-1}^n +U_j^{n+1} -U_{j-1}^{n+1} ) $$ which arises from the system $U'(t) ...
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39 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 ...
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3answers
312 views

Finite volume piecewise linear 2D advection develops instability

I'm developing a finite volume solver for the simple twodimensional advection equation with constant velocities $u, v$ and constant mesh spaces $\Delta x$: $$ \frac{\partial \rho}{\partial t} + u \...
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1answer
165 views

Advection equation in 2D using finite differences - the scheme works, but the pulse loses “energy”

I am trying to solve the following equation $ \partial_t g(x,y,t) = - v\left(\partial_x + \partial_y\right) g(x,y,t)$ using finite differences (here $v>0$). The equation is also solvable ...
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39 views

Understanding “characteristics”: texts/references and background needed?

I'm trying to get on top of "characteristics" as a way of solving fluid dynamic PDEs, coming from this question and using the textbook suggested there (Whitham, Linear And Non-linear Waves). I'm ...
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1answer
146 views

Is there a general analytic solution to 1D advection of velocity, $u_t=-uu_x$?

This is to help me relate continuous and discrete, predict what my scheme should be doing, and move toward using the method of manufactured solutions. There's a solution for constant velocity $c$ ...
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1answer
171 views

Solving the Advection Equation with Forcing using the Discontinuous Galerkin Method

I've been learning about the Discontinous Galkerin Method by reading the book by Hesthaven and Warburton and have ran into a problem with the advection equation with forcing $u_t + u_x = g(x,t)$ ...
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4answers
752 views

Advection equation with finite difference: importance of forward, backward or centered difference formula for the first derivative

I am a newbie in finite difference methods, so I apologize in advance if the question is trivial. I am trying to solve the advenction equation, i.e. $\frac{\partial \phi(x,t)}{\partial t} + v \frac{\...
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Extrapolating to non-fluid cells (for Shallow Water Equations), for a shore/beach?

The water height $h$ and 2d velocity field $(u,w)$ are "extrapolated to non-fluid-cells, i.e., setting $h$ equal to the value in the nearest fluid cell." [Bridson] I'm using finite differences. ...
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1answer
119 views

Finite difference method not working for advection PDE with negative coefficient

I'm trying to solve a very simple advection PDE $\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0$ where $c<0$. I have been able to implement a simple Modelica code to solve the ...
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148 views

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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3answers
605 views

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 ...
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121 views

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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1answer
302 views

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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82 views

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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1answer
82 views

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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80 views

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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3answers
143 views

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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180 views

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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1answer
151 views

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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2answers
110 views

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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133 views

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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1answer
239 views

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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198 views

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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189 views

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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1answer
536 views

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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1answer
354 views

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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1answer
189 views

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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1answer
218 views

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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2answers
652 views

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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2answers
145 views

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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123 views

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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1answer
2k views

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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1answer
132 views

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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2answers
387 views

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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1answer
178 views

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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2answers
202 views

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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1answer
2k views

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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2answers
771 views

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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183 views

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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1answer
220 views

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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2answers
203 views

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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2answers
441 views

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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1answer
10k views

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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1answer
499 views

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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1answer
103 views

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 ...