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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### 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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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. $\frac{\partial \phi(x,t)}{\partial t} + v \frac{\... 0answers 47 views ### Extrapolating to non-fluid cells (for Shallow Water Equations), for a shore/beach? The water height$h$and 2d velocity field$(u,w)$are "extrapolated to non-fluid-cells, i.e., setting$h$equal to the value in the nearest fluid cell." [Bridson] I'm using finite differences. ... 1answer 159 views ### Finite difference method not working for advection PDE with negative coefficient I'm trying to solve a very simple advection PDE$\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 234 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, ... 3answers 1k 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 185 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 462 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$u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$(... 0answers 93 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 $$M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}... 1answer 88 views ### Deposition model in laminar flow I have a chamber full with a fluid flowing horizontally in laminar regime from one side to the other. It carries a suspension with concentration c. This suspension also falls to the bottom of the ... 0answers 109 views ### Numerical diffusion in during advection of a free surface in an FE context I am currently working on a project where a two-phase flow is considered. The phases are described using a level set approach and a signed distance function from the interface between the phases where ... 3answers 189 views ### When is it safe to ignore the diffusion term in an advection-diffusion equation? Given the one dimensional equation: \epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0 with 0\le\epsilon \ll1 with boundary conditions u(0) = 0 and u(1) = 2, we ... 0answers 214 views ### Simple steady-state advection problem: do I need FVM with upwind scheme? I have a 2D (x,y) scalar advection problem that describes net blowing snow (q) transport at a point. This takes the form$$q = A - F*\nabla\cdot(q {\bf \hat u}),$$where A (kg\cdot m^{-2}\cdot s^{... 1answer 291 views ### Upwind difference for velocity in staggered grid I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf In the paper, the nonlinear term is treated as mix of central central difference and upwind difference using a ... 2answers 110 views ### How to support or contradict a hypothesis on unconditional stability using numerical optimization The main motivation behind my next question is that I think I derived a higher order numerical scheme for linear advection equation that is unconditionally stable using Von Neumann stability analysis. ... 0answers 183 views ### Spherical Advection Discretization (boundary nodes) Consider the spherical advection problem: describing the conservation of a property u in a closed spherical domain.$$ \frac{\partial u}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^... 0answers 65 views ### 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: $$\partial_{t}Q + \operatorname{... 1answer 434 views ### More Smearing with decreasing timestep in advection problems I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate). Let there be a equally ... 0answers 265 views ### Corner Transport Upwind for Linear Advection in Arbitrary Velocity Field I need to implement a 3D version of the Corner Transport Upwind (CTU) finite volume method (in python); and so I've been reading Leveque, "Finite Volume Methods for Hyperbolic Problems" which I think ... 0answers 235 views ### Numerical solution of non-linear advection equation other than inviscid burgers I am solving a non-linear advection equation of the form u_t + f(u)_x = 0 where f(u) is a complicated function of u. I am solving this equation using a first order fully implicit scheme (... 1answer 695 views ### Numerically computing the advection equation I am trying to write a program to compute the advection equation.$$u_t +u_x = 0$$I use the spectral method for the spatial derivative u_x and the leapfrog method for the time derivative u_t. ... 1answer 456 views ### Discretization method for advection equation without numerical diffusion Given the advection equation for an incompressible flow field$$\frac{\partial c}{\partial t} + \mathrm{Pe} \frac{\partial c}{\partial x} = 0$$what would the best method be for discretizing this ... 1answer 312 views ### boundary conditions of linear advection problem I am solving the 1D advection problem given by:$$\frac{\partial u}{\partial t}=-c\frac{\partial u}{\partial x}$$where c is the wave speed, and u is the unknown field variable, and x and t are time ... 1answer 308 views ### Find cfl condition We have the advection equation u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R} with initial condition u(0,x)=u_0(x). Suppose that we have the following sheme: I want to find the CFL ... 2answers 960 views ### Advection equation using the finite element method I want to solve this simple advection equation using the finite element method.$$\frac{dc}{dt}+v\cdot\nabla c = f$$What's the best FEM discretization for this? I have tried using the standard/... 2answers 150 views ### what do zero real parts of eigenvalues mean? Any good references? I am solving a 1D advection problem of the the form$$dQ/dt=[A]Q$$where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ... 0answers 143 views ### what do positive real parts of eigenvalues mean? I am solving a 1D advection problem of the the form$$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 ... 0answers 111 views ### 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 ... 1answer 3k 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{... 1answer 163 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) ... 2answers 467 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 ... 1answer 223 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 ... 2answers 360 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 ... 2answers 4k views ### 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 ... 1answer 3k 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}\,... 2answers 834 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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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 ...
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 \...