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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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52 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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22 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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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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39 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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39 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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45 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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46 views

Prove Lax entropy condition for conservation law

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
47 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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69 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 ...
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84 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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38 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
57 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
35 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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36 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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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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47 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 ...
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117 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
30 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 ...
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1answer
78 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
51 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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50 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\...
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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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12 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 ...
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36 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}}...
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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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35 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)...
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1answer
45 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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23 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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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
111 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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43 views

Non piecewise $C^1$ solutions of conservation laws

I studied the following theorem:(Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \rightarrow \mathbb{R} $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if ...
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1answer
54 views

Convexity and entropy for conservation laws

I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1). While defining the entropy why do we consider only convex flux? Can it be extended to general entropy flux? Please ...
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29 views

Solving a nonlinear system of equations where all the nonlinearities are on the diagonal?

Say you are given $c\in R^n$ and want to find $x \in R^n$ so that $x^TAx+Bx=c$, where A is diagonal and B is full. Is there some method which converges faster or has less computation time than the ...
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1answer
51 views

Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
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79 views

The Rankine-Hugoniot jump conditions for conservation and balance laws

Consider two problems: $$(1) \hspace{1cm} u_t+f(u)_x = 0, $$ $$(2) \hspace{1cm} u_t+f(u)_x = g(u). $$ Problem (1) represents system of conservation laws, and problem (2) represents system ...
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25 views

Definition of “timelike”

Michael Taylor defines timelike in his book Pseudodifferential Operators (1981, Princeton Legacy Library, Chapter 4, §4, Finite propagation speed) as follows: Let $L$ be a strictly hyperbolic ...
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1answer
33 views

An explicit form for a solution of a 1st order PDE

I was trying to solve this probleme using the method of differentials : $\begin{cases} \frac{\partial u}{\partial x} +u\frac{\partial u}{\partial y} = 0 ~~~~~~, (x,y)\in \mathbb{R}^{*}_{+} \\u(x,...
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1answer
136 views

Trying to understand hyperbolic canonical form transformation

I can't figure out what this author did in the text pictured below. Can someone PLEASE help. The author is transforming hyperbolic PDE's into canonical form. The author displays a general 2nd order ...
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1answer
88 views

Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation

I am trying to understand the intution behind use of energy estimate to prove existence and uniqueness(which is clear the energy estimate) of solution to hyperbolic equations. What is the basic idea ...
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1answer
31 views

Uniqueness of a solution to a 2nd order linear hyperbolic PDE

We proved in the lecture, that the weak solution of the linear hyperbolic equation \begin{align} u_{tt}+Lu&=f\ \ \ \mathrm{in}\ (0,T)\times\Omega \\ u &= 0\ \ \ \mathrm{on}\ (0,T)\times\...
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1answer
47 views

Numerical Solution to Advection Equation

I was wondering if classic Schemes like Lax-Friedrichs, Lax-Wendroff or Upwind Schemes work for the following PDE $$\dfrac{\partial u}{\partial t}+e^{-x}(\cos(t)+2)\dfrac{\partial u}{\partial x}=1+u^...
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1answer
66 views

PDE IVP : $zz_x + z_y = 0, \; \; z(x,0) = -3x$

Exercise : Given the PDE IVP : $$\begin{cases} zz_x + z_y = \quad0 \\ z(x,0) \; \; \; =-3x \end{cases}$$ a) Find the solution of it. b) Determine the lines of the $(x,y)$ plane on which the ...
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Proof request - Lemma (Non-continuous ODE solutions of $u_t + A(u)_x = 0$

Lemma - Protasis : Consider the PDE $u_t + a(u)u_x = 0$. Now, let $A'(u) = a(u)$. The PDE expression can be written as $u_t + A(u)_x = 0$. For this equation to hold, under the sense of ...
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49 views

ODE IVP : $u_t + uu_x = 0, \; \; u(x,0) = \cos \pi x$ [duplicate]

Exercise : Show that a smooth solution of the initial value problem $$\begin{cases} u_t + uu_x = 0 \\ u(x,0) = \cos \pi x \end{cases}$$ must satisfy $u = \cos[\pi(x-ut)]$. Also, show that when $...
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1answer
71 views

Crank-Nicolson for coupled PDE's

$\newcommand{\T}{T}$ $\newcommand{\partiald}[2]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\partialdd}[2]{\frac{\partial^2 #1}{\partial #2^2}}$ I am trying to solve a set of coupled PDE's with ...
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2answers
114 views

Why is the solution single-valued?

I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-...
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0answers
108 views

Local truncation error for Crank-Nicolson scheme for hyperbolic PDE $u_t+au_x = 0$

I am just trying to work out the LTE of the Crank-Nicolson scheme, however i do not get the same answers the book. here is my working if anyone could have a look and tell me what i am doing wrong, ...
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1answer
84 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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1answer
66 views

Find the weak solution of the conservation law

The picture above is the question I have been asked to do. For part ii) a) what I have done is that I have said that $g'(v) = v^{\frac{1}{2}}=u$ therefore it does satisfy the scalar conservation laws ...