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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### Concept of Hermite WENO scheme

Hermite WENO schemes (HWENO, paper2004 and paper2015) are said to be known extension of WENO schemes evolves slopes, so two variables $\{\overline{u}_i,\,\overline{v}_i\}$ are updated in time. ...
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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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### 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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### 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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### 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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### 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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### 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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### 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)...
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