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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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1answer
49 views

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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2answers
80 views

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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2answers
57 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
41 views

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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2answers
82 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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0answers
40 views

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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1answer
79 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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31 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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1answer
73 views

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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19 views

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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1answer
35 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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20 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
65 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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1answer
39 views

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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65 views

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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1answer
27 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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0answers
33 views

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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1answer
62 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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0answers
48 views

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
62 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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2answers
95 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 ...
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0answers
38 views

Definition of weak solution of a PDE that is given in the nondivergent form

Firstly, I would like to introduce two problems. A Riemann problem for a system of conservation laws given in divergent form: $$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)= \...
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29 views

Showing that the Riemann invariant $\frac{1}{2} (u^2+v^2) + \int \frac{c(p)^2}{p} dp $ is conserved along the characteristic $dy/dx = v/u$

I need to show that the Riemann invariant $R = \frac{1}{2} (u^2+v^2) + \int \frac{c(p)^2}{p} dp $ is conserved along the characteristic $dy/dx = v/u$. My system of equations are: \begin{aligned} (pu)...
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1answer
75 views

Characteristic curves for second-order Tricomi equation

Consider the Tricomi equation $$yu_{xx} + u_{yy} = 0$$ Find ordinary differential equations describing the real characteristic curves and solve these ODEs to obtain equations for the ...
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0answers
45 views

An estimate for a 1d hyperbolic PDE

Let $L, T, \lambda> 0$ be fixed, and let $f \in C^1([0,T];H^1(0,L))$, $g \in C^1([0,T];H^1(0,L)) \cap C^2([0,T];L^2(0,L))$ and $v^0 \in H^1(0,L)$. Consider the problem $$ \begin{cases} \partial_t ...
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1answer
50 views

Show that a solution of $u_t+(|u|^\alpha)_x=0$ violates entropy condition

Consider $$u_t+(|u|^\alpha)_x=0, \quad\alpha>1$$ Given the initial condition $$u(x,0)=\begin{cases} 0, x<0\\1,x>0\end{cases}$$ a) Find a solution for $u(x, t)$ that is continuous for ...
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1answer
59 views

Prove Lax entropy condition for conservation law with convex flux

A conservation law $u_t + \phi(u)_x = 0$ is considered. For a flux $\phi(u)$ satisfying $\phi'' (u) > 0$, show that the entropy condition in the form: $u(x + a, t) − u(x, t) \leq \frac{aE}{t}$, ...
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1answer
58 views

Rankine-Hugoniot jump condition for non-homogeneous conservation law

Consider the first order partial differential equation $$ \frac{\partial u}{\partial t} + 3u^2 \frac{\partial u}{\partial x} = -\alpha u , \tag{1} $$ where $\alpha>0$ is a constant. This ...
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2answers
77 views

Shock formation condition in IVP of $u_t + uu_x + \alpha u = 0$

Consider $u_t + uu_x + \alpha u = 0$ for $t > 0$, all $x$ where $\alpha > 0$ is a constant. Find the characteristic equations for the equation with initial data $u(x, 0) = f(x)$ given. Show ...
3
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1answer
149 views

An initial- and boundary-value problem for Burgers' equation with no solution

Prove that there is no solution to the following Cauchy problem: $$\begin{align} u_t+uu_x& =0&\quad x&\in(-1,1), t\gt0 \label{1}\tag{1}\\ u(x,0)&=x&\quad x&\in[-1,1] \label{2}\...
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39 views

Why a classical solution of the wave equation has to be $C^2$?

A (classical) solution of the wave equation $$ u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+, $$ is required to be of class $C^2$. Why? I mean, why one imposes that all second ...
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2answers
58 views

Non-linear partial differential equation with conditions

I have a pde with conditions, for which I'm looking for an analytical solution : $\partial_t f(t,x)+f(t,x)\partial_x f(t,x)=0$. $f(0,x)=0 \, , \, f(t,0)=0 $. $f(t,x)$ is defined over : $\mathbb{R}^+...
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1answer
44 views

How do I show that the system is hyperbolic if $u^2 + v^2 > c^2$

I know that for a system to be hyperbolic it must have 2 real distinct eigenvalues $\lambda$ where $\det(B-\lambda A)=0$. My system of equations are: \begin{aligned} (pu)_x + (pv)_y &= 0 \\ p(uu_x ...
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1answer
52 views

Method of characteristics for system of linear transport equations

If I have a system of pde $$\begin{cases} u_t+v_x=0\\ v_t+u_x=0\\ u(x,0)=u_0(x), v(x,0)=v_0(x)\end{cases}$$ how to extend the idea of method of characteristics to this situation? How do I ...
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0answers
14 views

Uniqueness of Riemann function for hyperbolic PDEs

Show that the Riemann function for a general hyperbolic PDE in canonical form is unique. That is, prove that the problem $$ R_{xy} - (aR)_x - (bR)_y + cR = 0 $$ with $$ \begin{aligned} &R_x = bR ...
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1answer
80 views

Why is the Lax entropy condition sufficient for uniqueness of solution for the shallow water equations?

Why is the Lax entropy condition $${\lambda _i}({{\mathbf{u}}_R}) \leqslant {\sigma _i} \leqslant {\lambda _i}({{\mathbf{u}}_L}),$$ where $i = 1,2$, a sufficient condition for uniqueness of the ...
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1answer
40 views

Hyperbolic system with nonhomogeneous boundary conditions

I want to solve this problem but I'm stuck in the last step. I have followed all the steps below, but I don't know how to finish. Any Ideas? We consider the standard wave equation with ...
6
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1answer
132 views

IVP for nonlinear PDE $u_t + \frac{1}{3}{u_x}^3 = -cu$

I'm trying to solve the following partial differential equations: $$ u_t + \frac{1}{3}{u_x}^3 = 0 \tag{a} $$ $$ u_t + \frac{1}{3}{u_x}^3 = -cu \tag{b} $$ with the initial value problem $$ u(x,0)=h(x)= ...
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1answer
34 views

Well balanced scheme

When do we say that a numerical scheme is well balanced? I could not find the precise definition. I read that these are the schemes which preserve steady state. But in which sense (like when do we say ...
2
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1answer
81 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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1answer
77 views

Determine the shock curve and sketch characteristics in $xt$-plane

Let $$\begin{equation} u(x,t)= \begin{cases} \frac{x-2}{t+2}&;x>\xi(t)\\0&;x<\xi(t)\end{cases} \end{equation}$$ be a weak solution to $u_t+(\frac{u^2}{2})_x=0$ in $\mathbb R\times(0,\...
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1answer
61 views

Sketch solution of IVP for nonconvex scalar conservation law

Compute explicitly the unique entropy solution of $u_t+(\frac{u^3}{3})_x=0$ in $\mathbb R\times(0,\infty)$, subject to \begin{equation} u(x,0)= \begin{cases} 0 &;x\le0\\ 2 &;0<x<2\...
3
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0answers
55 views

Uniqueness for classical solution of PDE

I have the following conservation law in my hand: $\partial_{t} u + \partial_{x}f(u) = -u$, with the associated initial data $u(x,0) = u_{0}(x)$ - where $u_{0} \in C^{1}$. I have to show ...
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0answers
14 views

principal symbol for hyperbolic operators on a real vector bundles

For a linear partial differential operator is elliptic if it's principal symbol is a linear-space isomorphism for all nonzero covector fields. I wonder if there is a similar determination for a linear ...
2
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0answers
37 views

Why does there exist Riemann invariants for this linear system of two equations?

If a linear system of two PDE’s $${\bf{A}}{{\bf{x}}_{{u_2}}} + {\bf{B}}{{\bf{x}}_{{u_1}}} = {\bf{0}}$$ (where ${\mathbf{x}} = {({x_1},{x_2})^T} \in {\mathbb{R}^2}$, and ${\mathbf{A}}$ and ${\mathbf{B}}...
3
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0answers
85 views

Why is the domain of dependence for a system of hyperbolic PDE’s an interval on the $x$ -axis?

I’m looking at the hyperbolic system ${{\mathbf{u}}_t} + {\mathbf{A}}({\mathbf{u}},x,t){{\mathbf{u}}_x} = {\mathbf{h}}({\mathbf{u}},x,t)$ $\quad$ (1) where ${\mathbf{u}}(x,t) \in {\mathbb{R}^n},\;\;...
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1answer
36 views

System of 2nd-order coupled PDE's

I have the following system of PDE's $$ B\big( -\partial_x^2 W^{00}+\partial_x \partial_t W^{01}\big) -2\beta_0 W^{00}=-\frac{\alpha_0}{2}\\ B\big(\partial_t^2 W^{00}-\partial_x \partial_t W^{01}\big)...
1
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1answer
46 views

Upwind Schemes meaning

What is an upwind scheme?(Why the name "upwind") Why Gudunov scheme for conservation laws is an upwind sceme?
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0answers
27 views

Hyperbolic Conservation Laws

Why the name Hyperbolic Conservation law for $u_t+f(u)_x=0$ Is there any parabolic or elliptic conservation laws?
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0answers
34 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 ...