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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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Transport equation 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
30 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
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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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51 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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9 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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30 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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1answer
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
34 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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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 ...
2
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1answer
52 views

weak solution of 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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38 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 ...
2
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1answer
36 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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27 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
46 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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50 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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0answers
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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
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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
63 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
27 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
45 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
60 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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41 views

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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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
61 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
99 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
56 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
69 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
44 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 ...
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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
41 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 ...
2
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1answer
79 views

Convergence of Lax-Wendroff method

What is the convergence condition of the Lax-Wendroff scheme to solve $U_t+AU_x=0$, where $$A=\begin{pmatrix}2&1&0 \\ 1&1&2\\0&2&-1\end{pmatrix}$$ I don't know what I should ...
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2answers
80 views

Is it possible to solve a hyperbolic moving boundary problem?

J. L. Davies says in his book, "The basic principle in PDEs is that boundary value problems are associated with elliptic equations while initial value problems, mixed problems, and problems with ...
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1answer
210 views

The method of Characteristics for Burgers' equation

I'm trying to solve numerically the inviscid Burgers' equation $u_t + u u_x = 0$ with the method of characteristics. Most of all, I want to see how the numerical solution gets "multiple-values" for ...
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1answer
78 views

Advection equation with discontinuous initial condition

I'm trying to solve in MatLab/Octave the advection equation, with $a > 0$ : $$\begin{cases} u_t + au_x=0\\ u(x,0)=u^{0}(x) \end{cases}$$ with $u^{0}(x)=\begin{cases} 1.5 \quad x<0 \\0.5 \quad ...
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0answers
76 views

How many boundary conditions for a system of PDEs?

Consider a system of hyperbolic PDE's, \begin{cases} \hat L_1 u_1(\vec x) = F\,,\\ \hat L_2 u_2(\vec x) = G\,, \end{cases} subject to the boundary conditions: \begin{cases} \hat B_1(u_1-u_2) = 0\,,\...
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1answer
27 views

Hyperbolic PDEs and Characteristic Curves

I refer to pp. 89-90 of "Partial Differential Equations of Mathematical Physics and Integral Equations" by Ronald B. Guenter & John W. Lee Given $u_{tt}=c^2u_{xx}$ for $x\in \mathbb{R}$ and $t>...
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2answers
402 views

Lax-Wendroff method for linear advection - Matlab code

$$ {\bf u}^{n+1} = {\bf u}^{n} - \frac{\Delta t}{2 \Delta x} {\bf c}.^*({\bf D}_{\bf x}{\bf u}^n) + \frac{\Delta t^2}{2 \Delta x^2} {\bf c}^2.^*({\bf D}_{\bf x x}{\bf u}^n) + \frac{\Delta t^2}{8 \...
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1answer
148 views

Help deriving Lax-Wendroff scheme for advection equation $u_t+c(x)u_x = 0$

Question 1: Consider the wave equation $$ u_t + c(x) u_x = 0 , $$ where $x\in \Omega \subset \Bbb R$ and $c(x)$ is a function of $x$. (a) Show that the Lax-Wendroff scheme for this PDE is ...
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1answer
303 views

Lax-Wendroff method for linear advection - Stability analysis

Question 1: Consider the wave equation $$ u_t + c(x) u_x = 0 , $$ where $x\in \Omega \subset \Bbb R$ and $c(x)$ is a function of $x$. (a) Show that the Lax-Wendroff scheme for this PDE is ...
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0answers
20 views

Order of accuracy, mixed $\Delta$ terms

I'm a bit confused about computing the order of accuracy for certain numerical methods for PDE (specifically Beam-Warming in this case, but generally as well). If I compute my local truncation error, ...
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2answers
92 views

Method of Characteristics for Variation of Burgers 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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2answers
66 views

Goursat Problem Solution

Show that the equation $u_{xy}+u_x=0$, $x\leq x_0$, $y\leq y_0$ has the solution $\alpha(x)e^{-y}+\beta(y)$ where $\alpha,\beta$ are arbitrary single variable functions. I have understood that I can ...
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0answers
18 views

Uniqueness and domain of dependence for wave equations.

Suppose there is a spacetime. If I have two functions $u_1$, $u_2$ satisfying one wave equation, and there is a subset $A$ such that $u_1,u_2$ coincide in $A$, then do we have that $u_1,u_2$ coincide ...
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1answer
34 views

Simple calculus

This problem is from nonlinear wave equation. From the following identity $$\partial_t\left(\displaystyle \frac{(\partial_t u)^2+|\nabla u|^2}{2}+\frac{u^6}{6}\right)-\operatorname{div}(\partial_t u\...
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27 views

Analytical solution to hyperbolic dirichlet problem with discontinuous initial condition

I have tried to look for a solution to the two dimensional wave equation (assume Cartesian or Polar setting) with Dirichlet boundaries and discontinuous initial conditions: $ u_{tt} = \Delta u \ \...
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0answers
18 views

Introduction to characteristic surfaces and bicharacteristics

I am currently studying the propagation of contact discontinuities in systems of hyperbolic PDE (multidimensional and transient). I have found that the concept of characteristics is helpful in ...
3
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2answers
694 views

Canonical form of PDE

I want to solve the following PDE using canonical form: $$xu_{xx} + 2x^2u_{xy} = u_x - 1$$ I have found the characteristic curves $\xi = y$ and $\eta = y-x^2$. By computing $u_{xx}$ and $u_{xy}$, I ...