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# Questions tagged [transport-equation]

For questions related to transport equations. The transport equation describes how a scalar quantity is transported in a space.

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### semilinear transport equation

Good morning everyone! I have to solve the following transport equation for the temperature $T(x,t)$: $$\frac{\partial T}{\partial t} + v \frac{\partial T}{\partial x} = k(T_a - T)$$ with boundary ...
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### Transport equation with dirichlet boundaries counditions

my fellows mathematicians, I have a question concerning the linear transport equation in a very simple case (1D with Dirichlet boundaries conditions) \begin{split} & f : x \in ...
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### Writing a PDE in a moving coordinate

I am trying to understand the following PDE by writing it in a reference frame marked by the red cross. $\partial_t h(x,t)+ \partial_x (u(x,t) h(x,t))=0,$ where h shows the height. I want to write an ...
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### Asymptotic analysis of a linear advection-diffusion equation

Consider the linear advection-diffusion equation for $t,x>0$ \frac{\partial c}{\partial t} + f(x)\frac{\partial c}{\partial x} = \varepsilon \frac{ \partial^2 c}{\partial x^2} \tag{...
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### Does transport equation preserve $L^\infty$ norm?

For transport equation with constant velocity $v$, $$u_t+v\cdot \nabla u=0, u(x,0)=f(x)$$ By method of characteristics, we have $u(x,t)=f(x-vt)$. Thus $$||u(x,t)||_{L^\infty}=||f||_{L^\infty}$$ i.e. ...
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### Solve transport equation with variable velocity $v(x,t)$

I want to get the solution of $$\frac{\partial u(x,t)}{\partial t}=v(x,t)\cdot \nabla u(x,t)$$ with $u(x,0)=f(x)$ The problem I encountered using method of characteristics is that: I cannot write the ...
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### Finite Difference Methods for 1D Transport Equation with Larger Stencil Widths

I'm attempting to re-create the numerical experiments in this paper for solving the 1D scalar advection (transport) equation$:$ $$\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0$$ ...
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### Making Sense of Method of Characteristics Solution Geometry

How can I make sense of the following surface as insight into understanding the method of characteristics? The partial differential equation this initially came from was $2xu_x + u_y = 0$, which has ...
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### Green's function of non-homogeneous advection equation

I am working on a physics problem which leads to an inhomogeneous PDE $$\left[ \frac{\partial}{\partial t} + \vec{v} \cdot \nabla_\vec{r} \right] f(\vec{r},\vec{p},t) = g(\vec{r},\vec{p},t)$$ which ...
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### Is the weak solution of the transport equation $\left(\frac\partial{\partial t}+b\cdot\nabla_x\right)u=0$, $u(0)=u_0$, still be given by $u_0(x-tb)$?

Let $d\in\mathbb N$, $b\in\mathbb R^d$ and $u_0:\mathbb R^d\to\mathbb R$ be differentiable. We can easily show that the unique (classical) solution of \begin{align}\left(\frac\partial{\partial t}+b\...
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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{\...
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### Prove a given function solves the transport equation PDE

From Evans' PDE, chapter 2.1.2 Prove $u(x,t)=g(x-tb)+\int_0^t f(x+(s-t)b,s)\,ds, u\in C^1(\mathbb R^n\times[0,\infty))$ solves \begin{cases} u_t+b\cdot Du=f &\text{in }\mathbb R^n\times(0,\infty),...
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### Fully-implicit scheme for reactive transport equation.

I am reading a paper by Zhang 2007 about reactive chemical transport where a fully implicit method was used in solving the transport equation given by, "At $\left(n+1\right)$th time step, ...
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### Energy estimate for a friedrich system with relaxation and periodic boundary conditions

I'm considering the following system of pdes $$\partial_t u + \sum\limits_{i=1}^3 \mathcal{A}_i\partial_{i} u = -\mathcal{R} u.$$ Where the matrices $\mathcal{A}_i$ are symmetrics, $u$ is valued in ...
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### Solving $u_t = (f(t)-1)u_x$

Good afternoon, Here's the question I'm currently working on: Determine the solution of the following transport equation: u_t = (f(t)-1)u_x \space \space \space ,\space x \in \mathbb{R}, \space 0 &...
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### Proper advection equation for non-conservative values in polar cylindrical coordinates.

I'm numerically solving a system of PDE's consiting of some conservation laws (Euler equations) along with advection of non-additive values (like molar mass or unit heat capacity, for example). The ...
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