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Questions tagged [hyperbolic-equations]

A hyperbolic PDE is a PDE that has a well-posedness initial value problem for the first $n-1$ derivatives. The Cauchy problem can be solved locally for arbitrary initial date along any non-characteristic hypersurface.

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Question on finite propagation speed of solution of hyperbolic partial differential equation

Consider the wave equation $u_{tt}-\sum a^{ij}u_{x_i}u_{x_j}=0$. Assume that for any $v\in \Bbb{R}^n$, we have $\sum\limits_{i,j} a^{ij}v_iv_j\leq c^2\sum\limits_{k=1}^n v_k^2$. Prove that if $u,v$ ...
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Proving that a hyperbolic PDE admits a weak solution

I've been given a hyperbolic differential equation. The question is the following: Prove that the above problem admits a weak solution (Hint: write an equation satisfied by $v=e^{\lambda t}u$, then ...
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Weak solutions of Riemann problem - Burgers' equation for $u_l<u_r$

I am studying the book "Numerical Methods for Conservation Laws" by R.J. LeVeque (Birkhäuser, 1992) and I am having a lot of dificulties at some Exercises. Exercise 3.6 p. 30. There are infinitely ...
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Prove that shock wave is weak solution of Burgers' equation (Riemann problem)

In math modeling studies, I need to prove that $$u(x,t)=\begin{cases}u_l\qquad x<st\\ u_r\qquad x>st\end{cases}$$ where $$s=(u_l+u_r)/2$$ is a weak solution for the Riemann problem of ...
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Solving a first order hyperbolic PDE in two variables

Question: Consider the system \begin{align} \frac{\partial u}{\partial x} + u \frac{\partial u}{\partial y} + 2\frac{\partial v}{\partial y} & = 1 \\ \frac{\partial v}{\partial x} + 2v\frac{\...
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Integrating factor in canonical form of second-order linear equations

In the hyperbolic PDE, I have ticked the part I do not understand. How do they get it to $v_s(r,s)= r-1 + C(s)e^{-r}$ in the canonical form process? In the textbook, it's said that they're using some ...
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Problem with solving hyperbolic PDE using canonical form

I'm stuck when im trying to solve this equation: $\frac{1-n}{2}xu_x+\frac{n-1}{2}u-2x^2u_{xx}+b n\cdot x\cdot y \cdot u_{xy}=0$, where $u=u(x,y)$ and $b$ and $n>0$ are constant parameters. ...
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Where is the solution uniquely determined by the data?

Question: Solve the PDE $$y\frac{\partial u^2}{\partial x^2} + (y-x)\frac{\partial u^2}{\partial x \partial y} -x \frac{\partial u^2}{\partial y^2} = \frac{y-x}{y+x}\bigg(\frac{\partial u}{\partial ...
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How to show that strictly hyperbolic implies strongly well posed?

Consider a first order system $\partial_t u = P(D)u$ with $P$ given by $$ P(\xi) = \sum_{k=1}^n iA_k\xi_k. $$ Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $\mathbb{R}^n$) and ...
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Solve one-dimensional form of Euler’s equations

This is a home work problem. Please find the problem in the image attachment. Problem : Consider the one-dimensional form of Euler's equations for isentropic flow and assume that pressure $p$ is ...
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Symmetrizability of shallow water equations

Consider the shallow water equation \begin{equation}h_t+(hu)_x=0\\ (hu)_t+\left(hu^2+\frac{g}{2}h^2 \right)_x=0 \end{equation} I want to know the entropy of this system? I understood that if their ...
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Writing 2D linear system of balance laws in compact form

I have three equations $$\rho \frac{\partial }{\partial t}v = \frac{\partial}{\partial x}\sigma_{21} + \frac{\partial}{\partial z}\sigma_{23}$$ $$\frac{\partial}{\partial t}\sigma_{21} = a\frac{\...
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Integral form of the conservation law $u_t+f(u)_x=0$

Consider the conservation law given by $$u_t+f(u)_x=0$$ We know that in general weak solutions are not smooth but are bounded in $L^{\infty}$ norm (they do not belong to Sobolev spaces). However ...
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47 views

Mathemathical model equation PDE

I am studying models in DPE an the professor give us this problem: $\begin{cases}u_t+au_x=f(x,t)\\ u(x,0)=g(x)\end{cases}$ I've studied the transport equation and the Burger's equation. About heat ...
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Lax-Wendroff theorem for hyperbolic systems

For scalar conservation laws convergence of a numerical scheme is governed by Lax-Wendroff theorem. Is there any such results for hyperbolic system of conservation laws? In other words, Given a ...
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Behavior of the solution to the inviscid Burgers' equation

Consider the inviscid Burgers' equation $u_t+uu_x=0$ with the initial condition $$u_0=\begin{cases} 0, & x<0\\ x, & 0\leq x \leq 1\\ 1, & x>1 \end{cases}$$ I tried to implement ...
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94 views

Finding weak solutions of conservation law $u_t + (u^4)_x = 0$

Consider the conservation law $$ u_t + (u^4)_x = 0, $$ (a) Find the solution at $t=1$ with the following initial condition: $$ u(x,0) = \left\lbrace\begin{aligned} &1 && x<0 \\ &...
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127 views

Conservation law $A_t + (A^{3/2})_x = 0$ for flood water wave

The flood wave in a river follows the conservation law $$ A_t + (A^{3/2})_x = 0 $$ where $A(x,t)$ is the cross sectional area of the water at the location $x$ and time $t$. A sudden heavy rain ...
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113 views

Find weak solution to Riemann problem for conservation law

Find the weak solution of the following conservation law $$ u_t + (u^2)_x = 0 $$ with the initial condition $$ u(x,0) = \left\lbrace \begin{aligned} &u_l & &\text{if } x < 0 ,\\ &...
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Energy Momentum Tensor in Polar Coordinates

Consider the Minkowksi wave equation on $2+1$ dimensions: $\partial_{tt}u - \Delta u = 0$. I know how to write the 3 by 3 = 9 components of the energy momentum tensor for this in the coordinates $x_0 =...
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Implement boundary conditions in finite-volume code for conservation laws

For the numerical solution of scalar hyperbolic conservation laws using finite volume schemes. In order to implement the boundary conditions and the numerical fluxes, make use of Ghost cells. ...
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50 views

Fixed point equation to solve Burgers' equation IVP

Using the equation $u \equiv u ( x , t ) = u _ { 0 } ( x - t u ( x , t ) )$ to compute $u \left( T , x _ { j } \right)$ for the Burgers equation, where the Burgers equation is $u _ { t } + \left( \...
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How to solve $u_t + u_x =0$ with $u_0(x) = \sin(\pi x)$ with Characteristics?

Consider the initial-boundary value problem (IBVP) for the convection equation \begin{array} { l } { u _ { t } + u _ { x } = 0 \quad x \in [ a ( t ) , b ( t ) ] , t \in [ 0 , T ] } \\ { u ( x , 0 ) = ...
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Discrete entropy inequality for hyperbolic system

We know that any conservative and consistent numerical scheme for the hyperbolic system $$U_t+F(U)_x=0$$ where $U:\mathbb{R}\times \mathbb{R}^+ \rightarrow \mathbb{R}^n$ converges to a weak solution. ...
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Solve wave equation with non-constant wave speed using method of characterstics?

I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations: $$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
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Solving an inhomogeneous Burgers' equation with the method of characteristics

I am trying to solve the PDE $$u_t+5uu_x=u,$$ subject the boundary condition $u(0,t)=e^{14t}.$ I first start by defining the set of characteristic equations, $$\frac{dt}{1}=\frac{dx}{5u}=\frac{du}{...
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37 views

Solving the Hyperbolic PDE $u_{xx}-2u_{xy}-3u_{yy}=0$

I'm trying to show that the general solution of the hyperbolic PDE, $$u_{xx}-2u_{xy}-3u_{yy}=0,$$ is $u(x,t)=F(3x+y)+G(x-y)$. I thought I could reduce the given PDE to form two ODEs. So far I have ...
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Shallow water equation entropy concept

I am a beginner in shallow water equation. I am interested in the equation $$h_t+(hu)_x=0$$ $$(hu)_t+(hu^2+\frac{1}{2}gh^2)_x=0$$ I have the following doubts 1)Weak solutions are not unique in ...
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Numerical convergence of Godunov scheme

Consider a conservation law $u_t+ \left(\frac{u^2}{2}\right)_x=0$ We know from Lax-Wendroff theorem that Godunov Scheme(in fact any monotone conservative and consistent) converges and the limit is ...
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Shock formation for conservation law with initial and boundary data

Suppose we have $$u_t + f(u) u_x = 0$$ where $t, x > 0$, and initial conditions $u(x,0) = C$, where $C>0$ is constant, and $u(0,t) = g(t)$, where $t>0$. We know the solution is $$u(x,t) = F(x-...
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Burgers equation with sinusoidal bump initial data

Suppose we have $u_t + uu_x = 0 $ with $$ \phi(x) = u(x,0) = \begin{cases} 0, && x \leq 0, x > 1 \\ \sin \pi x, && 0 < x \leq 1 \end{cases} $$ If we parametrize our curve with ...
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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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Showing scheme is consistent with PDE $v_t+v_x = 0$

For the advection equation $v_t + v_x = 0$, For practice, i want to show the scheme FTCS $$ u_i^{n+1} = u_i^n - \frac{\Delta t }{2 \Delta x } (u_{i+1}^n - u_{i-1}^n ) $$ is consistent and ...
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114 views

Weak solution for Burgers' equation

I have the following IVP: $$ u_t + u u_x = 0,\qquad u(x,0) = \left\lbrace \begin{aligned} &0 && \text{if } x<-3 \\ &0.5 && \text{if } {-3}<x<-2 \\ &1 &...
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Oleinik condition is equivalent to Entropy Condition (PDE)

Show that for weak solutions of \begin{aligned} &u_t + f(u)_x = 0 \quad\text{in}\quad \mathbb{R}\times (0,\infty) \\ &u=u_0 \quad\text{on}\quad t=0 \end{aligned} The entropy condition ...
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123 views

TVD numerical solution of Burgers' equation with sinusoidal initial data

I have solved Burgers' equation $u_t + u u_x = 0$ using a Total Variation Diminishing method for the initial conditions $u(x,0) = \sin(2\pi x)$. I have used forward Euler time-integration with CFL of ...
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34 views

Characteristics for nonhomogeneous wave equation $y_{tt}=y_{xx} + f$

Consider the initial- and boundary-value problem $$\eqalign{ & {y_{tt}} = {y_{xx}} + f(t,x){\text{ }}{\text{, (t}}{\text{,x)}} \in {\text{(0}}{\text{,}}\infty {\text{)}} \times {\text{(0}}{\text{...
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Solve advection equation $v_t + v_x = 1$ numerically with Matlab

Consider the advection equation $$ v_t + v_x = 1 $$ with initial condition $$ v(x,0) = \begin{cases} \sin^2 \pi (x-1), & x \in [1,2] \\ 0, & \text{otherwise} \end{cases}$$ Clearly, we ...
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Proving Finite Speed of Propagation using Energy Methods

So I'm working through a problem where I am trying to show that the wave equation in one dimension has a finite speed of propagation. Assuming that the initial data, $u(x,0)$ and $u_t(x,0)$ are ...
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42 views

Rarefaction solution to Riemann problem for $x/t=0$

Given Burgers' equation: $u_{t}+\frac{1}{2}(u^{2})_{x}=0$ with the initial condition: $$u(x,0) =\begin{cases} \displaystyle u_{l},\quad x <0 \\ \displaystyle u_{r},\quad x>0 \end{cases}$$ ...
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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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Transform of a wave equation to a hyperbolic system

We consider the wave equation $$y_{tt}=y_{xx}+a(t,x)y, \text{ x$\in$(0,1)}, t\in (0,\infty).$$ with Dirichlet boundary conditions. I want to transform this equation to a hyperbolic system of the form ...
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Traffic flow with Dirac-$\delta$ source (on ramp)

I have been trying to solve the traffic flow equation with a singular source ($D>0$ large): $$ \rho_t + f(\rho)_x = D\delta(x) $$ with the flux $f(\rho)=\rho(1-\rho)$ and the initial data $\rho(x,0)...
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32 views

Compensated Compactness And Conservation laws

I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with 1) I have been reading many books where its been written in differnet ways. So What ...
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Properties of the solution of conservation laws

I learnt that if $u\in L^{\infty}$ is a weak solution of the IVP $u_t +f(u)_x=0$ with $u(x,0)=g(x)$ $\forall x \in \mathbb{R}$ Then $\int_{\mathbb{R}}u(x,t)dx=\int_{\mathbb{R}}g(x)dx$ $\forall t>...
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111 views

Solve inviscid Burgers' equation with shock [closed]

Consider the initial value problem for the Burgers' equation: \begin{equation} \begin{cases}u_t+uu_x=0\\u(x,0)=\phi(x)\end{cases} \end{equation} where $$\phi(x)= \begin{cases} 2 & \text{if }...
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What are $\delta$-shock solutions?

I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $\eta : y \rightarrow \eta_y \in Prob(\mathbb{R^n})$ which satisfies $\partial_t(\eta_y, \lambda) +\...
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1answer
71 views

First-order quasilinear PDE system analysis

For $(t,r)\in[0,\infty)\times[0,1]$ let be the following PDE's system $$\dot{\vartheta}= w' +w\vartheta' $$ $$\dot{w}= \vartheta'+ww' $$ along with initial conditions (i.c.) $$w(0,r)=w_i(r)\qquad \...
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107 views

Exact vs approximate Riemann solvers

I am trying to understand numerical methods for conservation laws. I am confused with few terminologies. I have the following doubts. When do we say that a numerical scheme for a conservation law ...