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Questions tagged [transport-equation]

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Linear advection equation with coefficient given at data points

I am working on a problem to solve some particle population balances. In the analysis of some experiments I got an equation of this type: $$\frac{\partial u}{\partial y} - \alpha(x, y) \frac{\partial ...
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Exact solution of advection PDE with numerical scheme

Consider the advection pde $v_t + a v_x = 0$ and let $R = \frac{a \Delta t }{\Delta x}$. Then, the FTFS scheme is given by $$ u_k^{n+1} = u_k^n - R (u_{k+1}^n - u_k^n ) $$ and $u_k^n = u(k \Delta x,...
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53 views

Volume Preserving Flow after Variable Change of Transport Equation

Suppose that we have the transport equation with non-consant coefficients in divergence from $$\partial_t f(t,x,\xi) + \nabla_{x,\xi}\cdot (F(x,\xi) f(t,x,\xi)) = 0 \\ f(0,x,\xi) = f_0(x,\xi)$$ where ...
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2answers
54 views

On the transport equation $x \cdot \nabla u = |x|^2$

I have no clue about how to solve the following system \begin{equation*} \begin{cases} x\cdot\nabla u=|x|^2, \quad x\in\mathbb{R}^n, \\ u|_{x_1=1}=3x_n. \end{cases} \end{equation*} Study the ...
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2answers
60 views

Explicit solution to IVP of PDE $\rho_t = [\rho (1-\rho)]_x$

When trying to determine the density profile $\rho(t,x)$ of a system of particles I came across the PDE: $$\frac{\partial \rho}{\partial t}=\frac{\partial}{\partial x}\big(\rho (1-\rho)\big), \qquad\...
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1answer
125 views

Leapfrog scheme for linear advection equation

I have $v_t+v_x=0$ with initial condition $v(x,0)= \sin^2 \pi(x-1)$ for $ x \in [1,2]$. My goal is to find numerical solution for $x \in [0,8]$ using the Leapfrog scheme $$ u_k^{n+1} = u_k^{n-1} - \...
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1answer
102 views

Implementing Lax-Wendroff scheme for advection in matlab

Given the advection equation $v_t + v_x = 0$ with initial condition $u(x,0) = \sin^2 \pi(x-1) $ for $1 \leq x \leq 2$ and $0$ otherwise. Solve this PDE exactly and compare with numerical solution ...
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4answers
79 views

One-way wave equation IBVP

Plese help me to find the solution of te following equation. For values of $x$ in the interval $[-2,3]$ and $t>0$ we consider the one way wave equation $$u_t+u_x=0$$ with initial data \begin{...
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39 views

Smoothness of solution to transport equation IVP

Consider the following PDE (the unknown is $u\in C^1(\mathbb{R}^{n+1},\mathbb{R})$): $$\partial_tu(x,t)+\sum_{k=1}^{n} a_k\partial_k u(x,t)= cu(x,t)$$ $$u(x,0)=g(x)$$ where $g \in C(\mathbb{R}...
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3answers
33 views

Showing solution satisfies PDE (chain rule help)

We have $$u(x, t) = g(x-at, 0) + \int_0^t f(x+a(s-t), s)ds$$ and want to show it satisfies the PDE $$\partial_t u + a \cdot Du = f$$ The solution given goes as follows $$\partial_t u + a\cdot Du = - ...
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1answer
35 views

Basic question about a first-order linear equation

I am just learning PDE. My lecture notes say the following: Consider the IVP $$ \begin{cases} u_t + a u_x = 0 \\ u(x,0) = \phi(x) \end{cases} $$ where $a \in \mathbb{R}$. Our goal is to reduce this ...
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0answers
25 views

Finite-difference vs finite-volume schemes for conservation laws

As far as I know we don't use finite difference scheme for conservation law because solution of conservation law makes no sense pointwise as its only in $L^{\infty}$. But however we use finite ...
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1answer
66 views

Finding the time when the speed of discontinuity becomes time-dependent in traffic flow

I am trying to use the following conservation law: $$u_t+f(u)_x=0 \ \ \ \ \text{where} \ \ \ f(u)=u(1-u).$$ IC: $u(x,0)=\frac{1}{4}$ for BC: $u(0,t)=1$ for $t>0$. I found the solution ...
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46 views

Exact solution for two coupled non-homogeneous transport equations

I want to solve the following system $$\eqalign{ & {y_t} = -{y_x} + z{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr & {z_t} = {z_x} + y{\text{ in (0}}{\text{,T)}} \times {...
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1answer
113 views

Find weak form of linear transport equation

I am stuck on the following problem that says: a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$ b) Show ...
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0answers
101 views

Vorticity transport 2D using Lax-Wendroff 2 steps scheme

I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form. Vorticity transport Eq.: $$\zeta_t+u.\zeta_x + v.\...
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1answer
40 views

Generalized linear transport equation

I stumbled upon a transport equation of the form $$u_t(x,t)=u_x(x,t) + u_x(1,t).$$ Since I can write it in the form $u_t(x,t) = Lu(x,t)$ where L is some linear operator I thought that there must be ...
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1answer
129 views

Inhomogeneous linear transport equation

Let $(a,b)$ be a subintervall of $(0,1)$. We consider the nonhomogeneous transport equation $$\eqalign{ & {y_t}(t,x) + c{y_x}(t,x) = {1_{(a,b)}}(x)f(t){\text{ }}{\text{, }}\left( {{\text{t}}{\...
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0answers
34 views

Uniqueness in 1D transport equation .

I am trying to prove uniqueness in 1D transport equation $$ \theta_t-(H\theta)\theta_x=0\\ \theta(\cdot,0)=\theta_0 $$ being $Hf$ the Hilbert transform of $f$, $x,t\in\mathbb{R}$. To do that, I ...
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1answer
83 views

Transport equation $p_t - (xp)_x = 0$ for density of substance

So I'm trying to do the following: i) Solve $$p_t - (xp)_x = 0 \quad\text{for}\quad (t,x) \in (0, \infty ) \times \mathbb{R}$$ $$p(0,x) = {p_0}(x) \quad\text{for}\quad x \in \mathbb{R}, {p_0} \...
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0answers
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Optimisation in public transport drivers behaviour

I am trying to model an optimisation problem. Here is the setting: The driver has to stop at each of the seven stations of the trip (They are ordered linearly so he first comes across station 1, then ...
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0answers
20 views

Flux Limiter for 2D Discontinuous Galerkin FEM

I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve $$ \dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J}, $$ ...
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36 views

Find the solution to PDE

I can solve this transport equation $$u_t + cu_x = 0$$ where c is constant. But I can't solve this where c is $$\vec{c}(x)$$ and $$\vec{c}(x,t)$$ Thank you
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0answers
83 views

Heat equation with time-dependent transport term

$\bullet$ I have the following heat equation with a time-periodic transport term: $$\kappa u_{xx} - a \sin(\omega t)u_x = cu_t$$ I'm considering a 1D domain over $-l<x<l$. I'd like to be able ...
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1answer
112 views

Verification of solution to a transport equation

Given $T > 0$ and some smooth function $\psi: \mathbb{R} \times (0,T) \to \mathbb{R} $ with compact support, let $v$ be the solution of the transport equation $$ v_t + bv_x = \psi \qquad \text{ in ...
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1answer
101 views

Finite difference method for non-linear PDE

I'm trying to numerically solve the following fairly simple non-linear PDE: \begin{equation} \frac{\partial \omega}{\partial t}=-V_g(\omega)\frac{\partial \omega}{\partial z} \end{equation} I've ...
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42 views

Rankine-Hugoniot condition for non piecewise smooth solutions

I studied the following theorem:(Rankine-Hugoniot condition) Let u:ℝ×[0,+∞)→ℝ u : R × [ 0 , + ∞ ) → R be a piecewise $C^1$ function. Then u is a weak solution of the conservation law if and only if ...
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1answer
177 views

weak solution of linear transport (advection) equation

How to show that $g(x-at)$ is the weak solution of the initial value problem $$u_t+au_x=0$$ $$u(x,0)=g(x)$$ where $ g(x)\in L^{\infty}(\mathbb{R})$ Definition: $u$ is said to be the weak solution of ...
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0answers
145 views

Parallel transport in a cone vs cylinder

My vague understanding of parallel transport is that when the guassian curvature is zero for all points in an area enclosed by a loop then a vector moved around that loop remains unchanged as long as ...
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1answer
72 views

Solving Quasi-Linear Transport Equation with two shockwaves.

I want to solve the following Partial Differential Equation by using the method of characteristics. This is the transport equation. \begin{align}u_t - (1-2u)u_{x}&=0, &-\infty < x < \...
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1answer
111 views

How to show $|t-x|$ is a weak solution to advection equation

On the wikipedia page on Weak solutions they give the example that $|t-x|$ is a weak solution to the 1st order wave equation $u_t +u_x=0$. I tried to follow through the working but get stuck at $$\...
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25 views

Convergence of numerical method for linear advection equation

For the PDE $u_t+cu_x=0$, we have the following method: \begin{aligned} \tilde u_i &=u_i^n-\lambda(u_i^n-u_{i-1}^n) \\ u_i^{n+1} &= \frac{1}{2}(u_i^n+\tilde u_i) - \frac{\lambda}{2}(\tilde ...
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1answer
75 views

Verifying Solution to 3D Advection Equation

The problem I'm having is straightforward. The 3D advection equation is $\frac{\partial u}{\partial t} + \nabla \cdot{\vec{u}\vec{c}} =0$ for a constant $\vec{c}$ in this case. The solution to verify ...
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3answers
101 views

Transport equation $u_t + xu_x + u = 0$ with $u(x_0, 0) = \cos(x_0)$

I have been studying PDEs using Peter Olver's textbook. I have learnt how to solve equations such as $u_t + 2u_x = \sin(x)$ subject to an initial condition such as $u(0,x) = \sin x$. Letting $\epsilon ...
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2answers
52 views

quasi-linear PDE

I am given the following equation. $xu_x+yu_y=1$ with the conditions $u(x,y)=y$ for all $x^2+y^2=1$ What I got so far. I calculated the characteristics $x'(t)=x(t),x(0)=x_0\in S_1$ which are $x(t)...
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0answers
124 views

Maximum Principle for a Diffusion Equation

Let $u$ be defined on $\overline{U_T} = \overline{(U \times (0,T])}$ where $U$ is a bounded open set. Let $u$ solve $$ u_t + b(x,t) \cdot \nabla u= \Delta u $$ where $b$ is an arbitrary, bounded ...
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1answer
86 views

Solving Heat Equation with Transport Term [duplicate]

I'd like to solve this equation: $$ u_t = u_{xx} + u_x $$ for $t>0$ with initial condition: $$ u(0,x) = \cos (2 \pi x) $$ And $1$-periodic, i.e we have that $u(t,x) = u(t,x+1).$ I was going to ...
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2answers
180 views

Method of Characteristics for traffic flow equation

I've recently been studying for qualification exams for my master's program. I've run into a few problems that I'm stuck on and hope that I can get some help here. We consider the hyperbolic ...
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0answers
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Different values on LHS and RHS using Backward Euler and Crank-Nicolson

I was doing some simulations Fortran to solve two transport equations for soot. One for Mass Fraction and the second for Particle Number Density. The first has an order of magnitude of 10^(-5) and the ...
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1answer
204 views

Adams Bashforth transport/wave equation

I'm trying to write a program to solve the 1st order 1-D wave equation (transport equation) with given initial condition and periodic boundary conditions on the domain [0, 1]. I need to use 2nd order ...
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1answer
585 views

Solving transport equation with pseudo-spectral and finite volume methods in MATLAB

I have a linear transport equation $$ \dot{c}(x; t) + vc_x(x; t) = 0 \tag{1}$$ on an interval $[0; 2π]$ with $v = 1$, periodic boundary conditions and two different initial values $$c(x; 0) = \sin(x)$$...
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2answers
205 views

Linear Transport Equation - Non-constant Speed

I'm struggling with the following question. I'm fairly new to PDEs as this is a question from an introductory course. $$u_t + (x^2u)_x=0$$ $$u(x,0)=1$$ I know it's of the form linear transport ...
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1answer
371 views

Conservation law and entropy condition problem

Consider the scalar conservation quantity: $u_t+(f(u))_x=0$ with $f(u)=u-u^2$, and initial condition $u(x,0)=\begin{cases} c, x<0 \\ 1, 0<x<1 \\ 0, x>1\end{cases}$ I want to be able to ...
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1answer
248 views

Nonlinear transport equation solution breakdown

Suppose you had the non-linear transport equation $$u_t+uu_x=0$$ with initial data $$u(x,0)=g(x)=\begin{cases} 1, & x>1 \\-x, & -1\leq x< 0 \\x, & 0\leq x\leq 1\end{cases}$$ Solving ...
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1answer
76 views

Similarity method Transport equation

I'm trying to use the similarity method/dilation to solve the linear transport equation $u_t+cu_x=0$. I make the following coordinate transformation: $(x,t)\mapsto (z,s)$ I apply the transformation $...
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1answer
138 views

Use Transport Theorem to prove equality of angular acceleration vectors.

My goal is to show that $\frac{^Bd}{dt}\vec{\omega}^{B/N}$ = $\frac{^Nd}{dt}\vec{\omega}^{B/N}$, where $\vec{\omega}^{B/N}$ is an angulr-velocity vector representing the rotation of frame $B$ with ...
2
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1answer
54 views

How to solve $uu_t+u_x=2$ with the method of characteristics?

How to solve this partial differntial equation? $\begin{cases} uu_t+u_x=2 \\ u(0,x)=1+x \\ t\geq0 \end{cases}$ I tried to do the method of characteristics as follows: $\frac{\partial x}{\partial s}=...
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2answers
156 views

Inhomogenous nonlinear transport equation $u_t+uu_x = -Du$

We have the following setup: $$u_t+uu_x = -Du \\ u(x,0)=\sin x.$$ The question is to find the time $T_s$ of a first shock formation. So basically, I need to solve the equation using method of ...
5
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1answer
133 views

Derivation of the weak transport equation

Let $\rho$ be a density on some space $M$. For all practical purposes some subspace of $\mathbb R^n$. Let $v$ be a smooth vector field with flow $\Phi_t$. The transport equation, as far as I ...
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0answers
74 views

Estimate related to linear transport equation

Suppose the conditions: $1)$ $p \in [1,+\infty]$ , $2)$ $u_{0}(x) \in L^{p}(\mathbb{R^{N}})$, $3)$ $c + \mbox{div} b \in L^{1}([0,T];L^{q}_{\mbox{loc}}(\mathbb{R^{N}}))$, $b \in L^{1}([0,T];(L^{q}...