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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### 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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### 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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### 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: \frac{\...
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^{-... 2answers 13k 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? 1answer 79 views ### Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity 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 ... 0answers 54 views ### Comparison between stability and accuracy of various Finite Difference schemes Im 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 ... 0answers 51 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. Im asking if those stability ... 0answers 59 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)$... 1answer 101 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 $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$\\ C(x,t) = \... 1answer 142 views ### Why this error occurs in my code for Lax Wendroff? I want to implement the Lax Wendroff method for a non linear advection equation which is$$\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}... 1answer 111 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 $$$u(0,x) = u_{0}(x) = \begin{cases} 10^4 (0.1-... 1answer 90 views ### Finite Difference for Advection Equation With Source I'm trying to find a convergent finite difference scheme for the PDE \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \... 0answers 78 views ### discretizing advection equation with variable wave speed + stability I currently have a code that solves u_t+ cu_x=0 with periodic boundary conditions, and constant c (using an upwind method). I'm wondering how I would alter this code to solve something of the form ... 0answers 62 views ### numerical solution to pde on an ellipse Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by (a\cos(r), b\sin(r)). The problem is the ... 4answers 936 views ### Finite volume piecewise linear 2D advection develops instability I'm developing a finite volume solver for the simple twodimensional advection equation with constant velocities u, v and constant mesh spaces \Delta x:$$ \frac{\partial \rho}{\partial t} + u \... 1answer 220 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 ... 0answers 44 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 ... 1answer 148 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: $$A =\begin{pmatrix} 1-a\cfrac{\Delta t}{\Delta x} & 0 &... 1answer 1k views ### Implementing structured grid boundary conditions using NumPy arrays? I am making a toy code in Python to solve the advection equation$$u_t + cu_x = 0$$with, for example, periodic boundary conditions. Background information The numerical grid is specified like this: ... 0answers 54 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 ... 1answer 114 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{... 5answers 802 views ### 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 ... 1answer 163 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 ... 2answers 5k views ### 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 ... 0answers 77 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} ... 0answers 378 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 ... 2answers 5k views ### 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 ... 4answers 8k views ### 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 ... 1answer 454 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 ... 1answer 640 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 ... 1answer 313 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) ... 4answers 2k 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{\... 3answers 2k 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)$$Note that for c > 0, the solution is ... 0answers 54 views ### Extrapolating to non-fluid cells (for Shallow Water Equations), for a shore/beach? The water height h and 2d velocity field (u,w) are "extrapolated to non-fluid-cells, i.e., setting h equal to the value in the nearest fluid cell." [Bridson] I'm using finite differences. ... 1answer 242 views ### Finite difference method not working for advection PDE with negative coefficient I'm trying to solve a very simple advection PDE \frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0 where c<0. I have been able to implement a simple Modelica code to solve the ... 0answers 300 views ### Crank-Nicolson scheme in space for advection equation Consider the equation$$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$, for t,x\in\mathbb{R}. I'd like to solve this equation forward in space and backward in time, ... 0answers 268 views ### Why can I not solve the negative advection equation (backwards in time)? Suppose we have the negative, inhomogeneous advection equation:$$\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})$$... 1answer 700 views ### Crank-Nicolson method for inhomogeneous advection equation Suppose we have the inhomogeneous advection equation$$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$for u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R} (... 1answer 6k views ### Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)... 0answers 100 views ### Discontinuos Galerkin Method - inhomogeneous advection problem I'm currently trying to get into this topic. I've learned that the basic scheme for the advection problem (D_{x}u+a*D_{t}u=0) can be solved in a scheme like$$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}... 1answer 104 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 ... 0answers 116 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 ... 3answers 324 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 ... 2answers 114 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. ... 2answers 988 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$\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
I have a 2D (x,y) scalar advection problem that describes net blowing snow ($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^{...