Questions tagged [transport-equation]
The transport-equation tag has no usage guidance.
0
votes
0answers
14 views
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:
$$\frac{\partial u}{\partial y} - \alpha(x, y) \frac{\partial ...
1
vote
0answers
24 views
Exact solution of advection PDE with numerical scheme
Consider the advection pde $v_t + a v_x = 0$ and let $R = \frac{a \Delta t }{\Delta x}$. Then, the FTFS scheme is given by
$$ u_k^{n+1} = u_k^n - R (u_{k+1}^n - u_k^n ) $$
and $u_k^n = u(k \Delta x,...
0
votes
0answers
53 views
Volume Preserving Flow after Variable Change of Transport Equation
Suppose that we have the transport equation with non-consant coefficients in divergence from
$$\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)$$
where ...
1
vote
2answers
54 views
On the transport equation $x \cdot \nabla u = |x|^2$
I have no clue about how to solve the following system
\begin{equation*}
\begin{cases}
x\cdot\nabla u=|x|^2, \quad x\in\mathbb{R}^n, \\
u|_{x_1=1}=3x_n.
\end{cases}
\end{equation*}
Study the ...
2
votes
2answers
60 views
Explicit solution to IVP of PDE $\rho_t = [\rho (1-\rho)]_x$
When trying to determine the density profile $\rho(t,x)$ of a system of particles I came across the PDE:
$$\frac{\partial \rho}{\partial t}=\frac{\partial}{\partial x}\big(\rho (1-\rho)\big), \qquad\...
2
votes
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
$$ u_k^{n+1} = u_k^{n-1} - \...
0
votes
1answer
102 views
Implementing Lax-Wendroff scheme for advection in matlab
Given the advection equation $v_t + v_x = 0$
with initial condition $u(x,0) = \sin^2 \pi(x-1) $ for $1 \leq x \leq 2$ and $0$ otherwise. Solve this PDE exactly and compare with numerical solution ...
4
votes
4answers
79 views
One-way wave equation IBVP
Plese help me to find the solution of te following equation.
For values of $x$ in the interval $[-2,3]$ and $t>0$ we consider the one way wave equation $$u_t+u_x=0$$
with initial data \begin{...
0
votes
0answers
39 views
Smoothness of solution to transport equation IVP
Consider the following PDE (the unknown is $u\in C^1(\mathbb{R}^{n+1},\mathbb{R})$):
$$\partial_tu(x,t)+\sum_{k=1}^{n} a_k\partial_k u(x,t)= cu(x,t)$$
$$u(x,0)=g(x)$$
where $g \in C(\mathbb{R}...
0
votes
3answers
33 views
Showing solution satisfies PDE (chain rule help)
We have $$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 = - ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
0answers
46 views
Exact solution for two coupled non-homogeneous transport equations
I want to solve the following system
$$\eqalign{
& {y_t} = -{y_x} + z{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr
& {z_t} = {z_x} + y{\text{ in (0}}{\text{,T)}} \times {...
0
votes
1answer
113 views
Find weak form of linear transport equation
I am stuck on the following problem that says:
a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$
b) Show ...
2
votes
0answers
101 views
Vorticity transport 2D using Lax-Wendroff 2 steps scheme
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$\zeta_t+u.\zeta_x + v.\...
0
votes
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 ...
2
votes
1answer
129 views
Inhomogeneous linear transport equation
Let $(a,b)$ be a subintervall of $(0,1)$. We consider the nonhomogeneous transport equation
$$\eqalign{
& {y_t}(t,x) + c{y_x}(t,x) = {1_{(a,b)}}(x)f(t){\text{ }}{\text{, }}\left( {{\text{t}}{\...
1
vote
0answers
34 views
Uniqueness in 1D transport equation .
I am trying to prove uniqueness in 1D transport equation
$$
\theta_t-(H\theta)\theta_x=0\\
\theta(\cdot,0)=\theta_0
$$
being $Hf$ the Hilbert transform of $f$, $x,t\in\mathbb{R}$. To do that, I ...
2
votes
1answer
83 views
Transport equation $p_t - (xp)_x = 0$ for density of substance
So I'm trying to do the following:
i) Solve
$$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} \...
1
vote
0answers
20 views
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 ...
0
votes
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
$$
\dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J},
$$
...
0
votes
0answers
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
3
votes
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 ...
1
vote
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
$$
v_t + bv_x = \psi \qquad \text{ in ...
2
votes
1answer
101 views
Finite difference method for non-linear PDE
I'm trying to numerically solve the following fairly simple non-linear PDE:
\begin{equation}
\frac{\partial \omega}{\partial t}=-V_g(\omega)\frac{\partial \omega}{\partial z}
\end{equation}
I've ...
1
vote
0answers
42 views
Rankine-Hugoniot condition for non piecewise smooth solutions
I studied the following theorem:(Rankine-Hugoniot condition)
Let u:ℝ×[0,+∞)→ℝ
u
:
R
×
[
0
,
+
∞
)
→
R
be a piecewise $C^1$
function. Then u is a weak solution of the conservation law if and only if ...
2
votes
1answer
177 views
weak solution of linear transport (advection) equation
How to show that $g(x-at)$ is the weak solution of the initial value problem
$$u_t+au_x=0$$
$$u(x,0)=g(x)$$ where $ g(x)\in L^{\infty}(\mathbb{R})$
Definition: $u$ is said to be the weak solution of ...
1
vote
0answers
145 views
Parallel transport in a cone vs cylinder
My vague understanding of parallel transport is that when the guassian curvature is zero for all points in an area enclosed by a loop then a vector moved around that loop remains unchanged as long as ...
2
votes
1answer
72 views
Solving Quasi-Linear Transport Equation with two shockwaves.
I want to solve the following Partial Differential Equation by using the method of characteristics. This is the transport equation.
\begin{align}u_t - (1-2u)u_{x}&=0, &-\infty < x < \...
1
vote
1answer
111 views
How to show $|t-x|$ is a weak solution to advection equation
On the wikipedia page on Weak solutions they give the example that $|t-x|$ is a weak solution to the 1st order wave equation $u_t +u_x=0$.
I tried to follow through the working but get stuck at $$\...
0
votes
0answers
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 ...
2
votes
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 ...
4
votes
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 ...
0
votes
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)...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
0answers
74 views
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 ...
1
vote
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 ...
0
votes
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
$$c(x; 0) = \sin(x)$$...
2
votes
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 ...
3
votes
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 ...
5
votes
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 ...
1
vote
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 $...
0
votes
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
votes
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}=...
2
votes
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
votes
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 ...
0
votes
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}...
