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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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2 votes
1 answer
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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,...
Yoni Verhaegen's user avatar
6 votes
2 answers
694 views

What is the advantage of using a particular RK Scheme?

The Wikipedia article on Runge-Kutta Methods lists several examples of each order. My question is, are there any particular advantages using one particular scheme over another of the same order? I ...
Jacob Ivanov's user avatar
5 votes
1 answer
131 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 ...
vainia's user avatar
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2 votes
0 answers
48 views

Upwind scheme flux conservation not satisfied in 2D

I have read that the upwind scheme is flux conservative. Then by my understanding, in the absence of sinks/sources and with absorbing boundaries, the amount of the quantity leaving through the ...
ThreeOrangeOneRed's user avatar
0 votes
0 answers
61 views

Numerical algorithm to compute second order solution of non-linear advection with source

I would like to solve the following equation, $$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x}\left({\mathcal A}f\right)=S(x,t)$$ where $f$ is a function of $x$ and $t$, ${\mathcal A}$ is a ...
Sayan's user avatar
  • 97
3 votes
1 answer
183 views

Numerical scheme for the level set equation that solves inverse mean curvature flow problems

I am considering the problem of simulating the evolution of an interface given as a curve in 2D (or surface in 3D) that evolves according to a velocity specified at the interface of the form: $$\vec{v}...
B0bby31's user avatar
  • 33
3 votes
1 answer
186 views

Discontinuous Galerkin for transport equation with non-constant advection

This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time-independent transport equation of the form $$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\...
UserA's user avatar
  • 139
2 votes
1 answer
131 views

Numerical solution of an advection equation, $\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$, with finite volume

I was trying to solve the following equation numerically, $$\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$$ I adopted the Godunov approach for discretising the ...
Sayan's user avatar
  • 97
1 vote
1 answer
76 views

blown up solution for linear advection in upwind method with finite difference

I am going to solve this advection equation regarding the flow simulation of an energy tower \begin{equation} Y_t+ v(t) Y_x =0 \end{equation} with the following boundary conditions which depend on ...
TMW's user avatar
  • 11
2 votes
3 answers
171 views

Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

I'm dealing with the numerical resolution of advection-diffusion-reaction equations. I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations. I'm ...
Max_89's user avatar
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4 votes
3 answers
530 views

Why not use the convolution theorem for explicit timestepping?

Consider the advection equation \begin{equation} \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = 0 \end{equation} I want to do a forward time, center ...
nalzok's user avatar
  • 181
0 votes
1 answer
123 views

WENO scheme on the advection of a fluid in a compressible porous media

I am working on reactive transport and I need to solve this advection equation: \begin{align} \frac{\partial (\phi C)}{\partial t} = - \nabla \cdot \big(\vec{q}(\phi) C\big) \end{align} with $\phi$ ...
Iddingsite's user avatar
1 vote
1 answer
98 views

Computing Jacobian in WENO scheme for advection in a porous media

I am trying to implement an advection equation for a coupled system of a two-phase flow in a porous media using a WENO scheme [1]. My equation is of the form: \begin{align} \frac{\partial (\phi(x,t) C(...
Iddingsite's user avatar
2 votes
2 answers
761 views

WENO Scheme for 1D linear advection equation

I have been stuck for a while now on a WENO scheme implementation and I have a few questions concerning the implementation, but also the theory behind it. For context: I'm currently working on ...
prickly's user avatar
  • 71
1 vote
0 answers
46 views

Semi-lagrangian method for compressible fluid (non divergence free)

I am looking for a semi-Lagrangian method for advection with a non divergence free velocity field. The equation is \begin{align} \frac{\partial C(x,t)}{\partial t} &= - \nabla \cdot (\vec{v}(x,t) ...
Iddingsite's user avatar
1 vote
0 answers
187 views

Solving 1D Advection Equation Using Midpoint-Rule and Finite-Differences

I want to solve the advection equation ($v_0 \in \mathbb R$) $$ \frac{\partial f}{\partial t} + v_0 \frac{\partial f}{\partial x} = 0 $$ using 2nd order Runge Kutta like the midpoint rule for the time ...
xotix's user avatar
  • 241
1 vote
0 answers
90 views

Chemical advection of a fluid in a porous media

I am trying to solve an equation for chemical advection of a fluid in a porous media with this equation: $\begin{align} \frac{\partial (\phi(x,t) C(x,t))}{\partial t} = - \nabla . (\vec{q(\phi, t, x)} ...
Iddingsite's user avatar
4 votes
1 answer
219 views

Stepping over a rapid oscillation in advection

As a model of a more complex problem, I am studying a linear advection equation with a rapidly oscillating velocity $$ \partial_t u + a \cos(\omega t) \partial_x u = 0 $$ with initial condition $u(x,0)...
Endulum's user avatar
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1 vote
0 answers
139 views

Attempt on 2d Advection with FDM - With Code

I tried to implement the 2d advection problem with a velocity field, that is not constant in space. My problem is, that the "mass" of my shifted density gets "eroded" or just ...
dba's user avatar
  • 295
4 votes
0 answers
85 views

Global reconstruction defined elementwise in a-posteriori error estimator

This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is ...
FEGirl's user avatar
  • 405
2 votes
1 answer
121 views

Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial

I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve $$\operatorname{div}(\beta u) = 0$$ $$u = g_D \text{...
FEGirl's user avatar
  • 405
4 votes
1 answer
253 views

Discontinuous Galerkin: confusion about the weak formulation for linear advection equation

In an introduction to Discontinuous galerkin methods, I have some problems in checking the weak formulation. I'm looking at page 16 here The context is the advection reaction equation: $$\operatorname{...
FEGirl's user avatar
  • 405
3 votes
0 answers
208 views

TVD Lax-Wendroff with non-constant velocity

I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D. The equation is the following: \begin{equation} \frac{\...
Iddingsite's user avatar
0 votes
0 answers
97 views

Transient advection equation with stabilized FEM

I am interested in solving the transient advection equation $\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
balborian's user avatar
  • 601
4 votes
1 answer
133 views

Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

\begin{equation} \begin{aligned} \frac{\partial N}{\partial t} &+ \frac{\partial J}{\partial r} = 0, \\ \frac{\partial N}{\partial t} &+ \frac{\partial }{\partial r}(N \upsilon ...
Ronnie1993's user avatar
1 vote
0 answers
128 views

Comparison between stability and accuracy of various Finite Difference schemes

I`m Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy. I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
Abdelrahman Mabrouk's user avatar
1 vote
0 answers
59 views

Stability plot of upward difference implicit time

I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number. I`m asking if those stability ...
Abdelrahman Mabrouk's user avatar
1 vote
0 answers
170 views

Integrating a wavelike equation with absorbing boundary conditions

I am trying to numerically solve the following equation: $\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$ ...
Joel's user avatar
  • 111
-1 votes
1 answer
230 views

Numerical solution of the advection equation with Crank–Nicolson finite difference method

I need to implement a numerical scheme for the solution of the one-dimensional advection equation $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$ \\ C(x,t) = \...
Winabryr's user avatar
0 votes
1 answer
300 views

Why this error occurs in my code for Lax Wendroff?

I want to implement the Lax Wendroff method for a non linear advection equation which is $$\frac{u_{i}^{n+1}-u_{i}^{n}}{t} + \frac{f(u_{i+1}^{n})-f(u_{i-1}^{n}) }{2h} -\frac{t}{2h} \left( F_{i+1/2}^{n}...
user avatar
1 vote
1 answer
694 views

What is wrong in the code for this upwind method?

I want to implement the upwind method in following advection equation problem : $$ u_{t}+2 u_{x} =0 ,$$ for $0\leq t \leq 1,$ $0\leq x \leq 1 $ $$ u(0,x) = u_{0}(x) = \begin{cases} 10^4 (0.1-...
user avatar
1 vote
1 answer
169 views

Finite Difference for Advection Equation With Source

I'm trying to find a convergent finite difference scheme for the PDE \begin{equation} \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \...
Mike D's user avatar
  • 141
0 votes
0 answers
201 views

discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
lrs417's user avatar
  • 11
0 votes
0 answers
82 views

numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
lrs417's user avatar
  • 11
2 votes
1 answer
321 views

Order of Accuracy Measurements on 1D Advection Methods

I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D. I am am currently testing the theoretical predictions of the order of ...
mivkov's user avatar
  • 203
1 vote
0 answers
49 views

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 ...
Natasha's user avatar
  • 433
2 votes
1 answer
395 views

Upwind finite difference: Matrix Implementation

I want to implement the upwind finite difference scheme for the 1D linear advection equation using a finite difference matrix in python: $$ A =\begin{pmatrix} 1-a\cfrac{\Delta t}{\Delta x} & 0 &...
ABCCHEM's user avatar
  • 21
1 vote
1 answer
2k views

Implementing structured grid boundary conditions using NumPy arrays?

I am making a toy code in Python to solve the advection equation $$u_t + cu_x = 0$$ with, for example, periodic boundary conditions. Background information The numerical grid is specified like this: ...
EssentialAnonymity's user avatar
0 votes
0 answers
114 views

Is semi-Lagrangian 1D advection identical to upwind Euler, when $|u|<\Delta x/\Delta t$?

This looked true to me, I worked through the algebra for 1D and confirmed it. But no one seems to mention it, so maybe I'm missing something... I'm using Bridson's SIGGRAPH 2007 course notes. [5MB ...
hyperpallium's user avatar
1 vote
1 answer
637 views

How to calculate the analytical solution of linear advection equation with Dirichlet's boundary conditions?

I am trying to find the solution of linear advection equation of the form: $\frac {\partial c}{\partial t}+u\frac {\partial c}{\partial x}=0$ $c(x,0)=0$ $c(0,t)=\{c_0 \ \text{for}\ t \leq t_1 \text{...
Adnan Hayat Khan's user avatar
2 votes
1 answer
215 views

Why the numerical solution of advection-dominant problem is challenging

In many CFD text books, usually there is a dedicated chapter for advection term discretization. Why discretization of such term in advection-dominated problems and near the discontinuities is ...
Rik's user avatar
  • 21
2 votes
0 answers
81 views

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} ...
Herman Jaramillo's user avatar
1 vote
0 answers
565 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 ...
GGG's user avatar
  • 173
2 votes
4 answers
1k 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 \...
mivkov's user avatar
  • 203
1 vote
1 answer
630 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 ...
Michele Cotrufo's user avatar
1 vote
1 answer
853 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 $c$ ...
hyperpallium's user avatar
7 votes
1 answer
369 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 $u_t + u_x = g(x,t)$ ...
user3209427's user avatar
6 votes
4 answers
3k 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{\...
Michele Cotrufo's user avatar
1 vote
0 answers
78 views

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. ...
hyperpallium's user avatar
2 votes
1 answer
309 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 ...
Foad's user avatar
  • 147