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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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 ...
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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:
$$\vec{v}...
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Discontinuous Galerkin for transport equation with non-constant advection
This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time-independent transport equation of the form
$$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\...
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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 ...
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Solving compressible euler equations in non-conservative form
I am trying to solve the following compressible 1D system of equations in two non-conservative form PDEs for $\rho, Y$ and one ODE for $v$
\begin{align}
\rho_t+(v+Q)\rho_x &= -\rho q(Y,T,\rho)-...
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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 ...
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3
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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 ...
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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 ...
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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 $\phi$ ...
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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(...
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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 ...
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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) ...
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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 ...
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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)} ...
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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)...
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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 ...
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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 ...
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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
$$\operatorname{div}(\beta u) = 0$$
$$u = g_D \text{...
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Discontinuous Galerkin: confusion about the weak formulation for linear advection equation
In an introduction to Discontinuous galerkin methods, I have some problems in checking the weak formulation. I'm looking at page 16 here
The context is the advection reaction equation: $$\operatorname{...
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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{\...
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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^{-...
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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 ...
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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 ...
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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 ...
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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:
$\phi(x, 0) = I(x)$
...
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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
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
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Why this error occurs in my code for Lax Wendroff?
I want to implement the Lax Wendroff method for a non linear advection equation which is
$$\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}...
user37062
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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 $
$$ u(0,x) = u_{0}(x) = \begin{cases}
10^4 (0.1-...
user37062
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Finite Difference for Advection Equation With Source
I'm trying to find a convergent finite difference scheme for the PDE
\begin{equation}
\begin{split}
u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\
u(x,0) &= 1 \\
u(1,t) &= 1. \\
\...
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discretizing advection equation with variable wave speed + stability
I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form
...
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numerical solution to pde on an ellipse
Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
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Order of Accuracy Measurements on 1D Advection Methods
I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D.
I am am currently testing the theoretical predictions of the order of ...
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Comparison of convection time - theoretical value vs computed
This is a follow up to my previous post here,
I'm solving for convection in 1D
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
The discretization of the above equation is ...
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1
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Upwind finite difference: Matrix Implementation
I want to implement the upwind finite difference scheme for the 1D linear advection equation using a finite difference matrix in python:
$$ A =\begin{pmatrix}
1-a\cfrac{\Delta t}{\Delta x} & 0 &...
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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:
...
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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 ...
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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{...
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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 ...
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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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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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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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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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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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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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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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votes
1
answer
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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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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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