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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43 views

Solve the equation $xy=x+2y+2009$ in integers

I know that the left side is a hyperbola and the right hand side is a line. So they have at most 2 solutions. I set $xy=k$ and solved for $y$, and after that substituted it on the right side. The ...
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40 views

Why is Godunov's Method considered expensive?

I am studying approximate Riemann solvers and often the motivation is that Godunov's method can be expensive to carry out, but I cannot see why. Say we have $$u_t + [f(u)]_x = 0$$ To my understanding, ...
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33 views

Explicit worked example of symmetrizing system of conservation laws

Pg 4 of this book gives the Euler equations for an ideal gas in the variables $(p, \textbf{v}, S)$ read: $$\frac{Dp}{Dt} + \gamma p \, \text{div} \textbf{v} = 0$$ $$\rho \frac{D\textbf{v}}{Dt} + \...
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1answer
49 views

Riemann Problem for Linear Hyperbolic Systems

I am following LeVeque's text "Numerical Methods for Conservation Laws" so I will be following his notation. Suppose we are solving $$u_t + Au_x = 0.$$ where $A \in \mathbb{R}^{m \times m}$. ...
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1answer
70 views

Overview of Godunov's Method

I am studying Godunov's method and I am unclear on a few details, it seems there are not many resources available regarding this method (at least to my knowledge). Suppose we have: $$v_t + [f(v)]_x = ...
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Solving a system of hyperbolic, elliptic and parabolic PDEs

$\text{Question: }$ Consider the following system for $u(x, y)$ $$ \begin{array}{l} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial x \partial y}-2 \frac{\partial^{2} u}{\partial ...
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1answer
171 views

Non-linear coupled PDEs $(\alpha_y/\beta)_y = K\alpha\beta = -(\beta_x/\alpha)_x$

Studying an engineering problem I came up with the following system $$ \begin{align} \frac{\partial}{\partial y}\left(\frac{1}{\beta}\frac{\partial\alpha}{\partial y}\right)&=K\:\alpha\beta\\ \...
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48 views

Method of characteristics in hyperbolic PDE's

I was reading Hyperbolic Partial Differential equations and its solutions through the method of characteristic. It stated that to find the solution, you need the characteristics $f$ and $g$ to ...
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84 views

Prove that when $|x|\geq M_1+M_0t, u(x,t)=0$.

For the following Cauchy problem: \begin{cases} \partial^2_tu-a^2(x,t)\partial^2_xu=f(x,t), x\in\mathbb{R},t>0\\u(x,0)=\varphi(x), \partial_t u(x,0)=\psi(x), x\in\mathbb{R},\end{cases} where \begin{...
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37 views

how to tell the types of higher order partial differential equations?

I'm reading P42 on Robert C. Rogers' book "An introduction to partial differential equations" , and I found the way he classifies higher order PDEs a little confusing, especially when it ...
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1answer
53 views

Are Riemann invariants & eigenspaces uniquely defined?

I am trying to understand some derivations surrounding the Riemann Invariants for the system: $$ \begin{pmatrix} u_t \\ \eta_t \end{pmatrix} + \begin{pmatrix} u & 1 \\ \eta & u \end{...
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1answer
41 views

Local existence and uniqueness and uniform a priori estimate for the C1 norm of the solution imply existence and uniqueness of global solution

In my book the "Global classical solutions for quasilinear hyperbolic systems" written by Li Ta-Tsien, following are written, Local existence and uniqueness of $C^1$ function+ uniform a ...
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67 views

TVD Lax-Wendroff with non-constant velocity

I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D. The equation is the following: \begin{equation} \frac{\...
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1answer
78 views

Convergence of numerical method for Burgers' equation

This is a classic question from Leveque's book Numerical Methods for Conservation Laws (Exercise 12.4 p. 135). Consider the 1-dimensional Burgers equation $u_{t}+(\frac{u^{2}}{2})_{x}=0$ with the ...
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1answer
50 views

Solving a second order PDE already in canonical form

I am trying to solve the second order PDE, $$\partial_x \partial_y u + \frac{1}{x} \partial_y u - y^2 = 0.$$ My problem is this is in canonical form and I don't see how to apply the method of ...
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66 views

Later shocks in Riemann problem [duplicate]

I am following section 3.4 of Evan's book "Partial Differential Equations". In this section he considers the Burgers equation, $$u_t + \bigl(\frac{u^2}{2}\bigr)_x = 0$$ with initial data $$g(...
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190 views

'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem. Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
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1 dimensional flux form semi lagrangian scheme for advection

I need to solve a conservative 1-dimensional advection equation of the following kind, $$\frac{\partial f}{\partial t}=\frac{\partial}{\partial x}\left(vf\right)$$ where, $f(x,t)$ is some function, $v$...
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Understanding the fast sweeping algorithm for Eikonal equations

My question comes from the paper "A Fast Sweeping Method For Eikonal Equations" by HongKai Zhao. Specifically, I have no clue why the first inequality stated after the statement of Theorem 3....
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Riemann invariants of a special hyperbolic system

How can one compute the Riemann invariants of the following one dimensional hyperbolic system? $$\begin{pmatrix} u \\ v \end{pmatrix}_t + \begin{pmatrix} -v & -u \\ |v|-k & \mathrm{sgn}(v)u \...
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Applying numerical schemes like Runge Kutta to hyperbolic PDEs where the partial derivative of one variable is a function of another variable

If anyone can help with how to begin in this case, it would be a great help. Imagine if we have a 1D hyperbolic system of equations like $\frac{\partial U}{\partial t}=-i\omega\zeta$ and $\frac{\...
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Rarefaction wave is an entropic solution!!

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly convex function. And $u:\mathbb{R}×[0,+\infty[ \rightarrow \mathbb{R}$ where $u(x,0)=u_l$ if $x \lt 0$ and $u(x,0)=u_r$ if $x \gt 0$ And ...
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1answer
52 views

solving partial differential equations second order

I have a problem with solution this hyperbolic equation: $$u_{xx}-u_{xy}-6u_{yy}=y\cos(x).$$ I found two characteristics: $$C_1=y-2x$$ $$C_2=y+3x.$$ I know that $u(x,y)=F(y-2x)+G(y+3x)-\sin(x)-y\cos(x)...
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1answer
153 views

Dam break problem for shallow water equations

This problem concerns the shallow water equations $\partial_tu+u\partial_xu+g\partial_xh=0$ $\partial_th+u\partial_xh+h\partial_xu=0$. where $g$ is a constant. Water of depth $h_0$ is at rest for $x &...
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Existence from Energy estimate for hyperbolic equation

Consider a Cauchy problem $$ Lu(t,x)=0, \quad u(0,x)=g(x), \quad(t,x)\in[0,T]\times \mathbb{R}^n $$ for a symmetric linear hyperbolic operator $L$. Suppose that we can prove an energy estimate of ...
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50 views

CFL Condition for hyperbolic balance laws?

Given is the scalar (hyperbolic) conservation law $$ \partial_t \rho(x, t) + \partial_x f(\rho(x,t)) = 0, $$ where $\rho : \mathbb{R} \times [0, \infty) \to \mathbb{R}$ and $f : \mathbb{R} \to \mathbb{...
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36 views

Discrete entropy inequality for scalar conservation laws

Consider a scalar conservation law $u_t+f(u)_x=0.$ A three point monotone scheme given by, \begin{eqnarray} u_i^{n+1}=u_i^{n}-\lambda (F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_i^n)) \end{eqnarray} where $F(u,...
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21 views

Putting an Hyperbolic partial differential equation in Canonical form

I am trying to show that if we have an hyperbolic equation of the form $au_{xx}+2bu_{xy}+cu_{yy}+du_x+eu_y+f=0$ then there exists a change of coordinates $\xi=ay-(b+\sqrt{b^2-ac})x ,\eta=ay-(b-\sqrt{b^...
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1answer
72 views

Riemann problem for a linear system

I want to know how I can determine the type of discontinuity (shock, rarefaction) for a Riemann problem of a linear system such as $$ q_t+Aq_x=0 . $$ Let's take the following example: \begin{align} A=\...
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1answer
42 views

Riemann problem for general Scalar Conservation Law [closed]

The Riemann problem for Burgers' equation $u_t +(f(u))_x = 0$, where $f(u)=\frac{1}{2}u^2$, has a shock solution: $$ u(x,t) = \left\lbrace \begin{aligned} &u_L &&\text{if}\quad x<st \\ ...
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1answer
143 views

Riemann problem for Burgers' equation with both shock waves and rarefaction waves.

Given the inviscid Burgers' equation with piecewise initial data $$ u_t + u u_x = 0,\qquad u(x,0) = \left\lbrace \begin{aligned} &0 && \text{if } x<-1 \\ &2 && \text{...
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1answer
80 views

Method of characteristics hyperbolic PDE

Consider the hyperbolic PDE : $$2u_{xx} + 8u_{xy} + 6u_{yy} = 0.$$ It can be shown using the method of characteristics that the above PDE has the following general solution: $$u(x,y) = F(y-x) + G(3x-y)...
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32 views

Existence of weak solutions to initial value problems of second-order hyperbolic differential equations on unbounded domains?

Evans gives a theorem for the existence of weak solutions for second-order hyperbolic differential equations. Specifically, he says if $U$ is an open, bounded subset of $\mathbb{R}^n$, we call $U_T = (...
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2answers
62 views

Numerical method to solve PDE system $\mathbf{A}\dot{\mathbf{U}} + \mathbf{B}\mathbf{U}^{'} + F(\mathbf{U}) = 0$

I would like to numerically solve a system of PDEs of the form $$\mathbf{A}\dot{\mathbf{U}} + \mathbf{B}\mathbf{U}^{'} + F(\mathbf{U}) = 0$$ Where, $\dot{\mathbf{U}}(s, t)$ represents the time ...
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1answer
130 views

Method of characteristics for $u_t + (1-2u) u_x = 0$ and shocks

My teacher just went over the method of characteristics and we did an example with shocks then drew a picture, but I wanted to clarify some things for myself. The example was $$u_t + (1-2u) u_x = 0$$ ...
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51 views

Existence and uniqueness of solution for a system of nonlinear pdes

Consider the system $$(I)\begin{cases} \frac{\partial u}{\partial t} = \nabla \cdot \left( D_1(x,t,u) \nabla u \right) + f_1(u,v) \\ \frac{\partial v}{\partial t} = \nabla \cdot \left( D_2(x,t,v)\...
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1answer
56 views

Second Order Methods for conservation laws

Consider the scalar conservation law $u_t+f(u)_x=0.$ I understand that Lax-Wendroff scheme is second order accurate, because of the way it is derived using Taylor series. However since this scheme is ...
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118 views

How to show instability of downwind and centred scheme

Good day :) I'm trying to show for the advection equation $q_t+uq_x=0$ with $u\geq 0$, for which the downwind scheme $Q_i^{n+1} = Q_i^n-\frac{u\Delta t}{\Delta x}(Q_{i+1}^n-Q_i^n)$ centred scheme $...
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1answer
83 views

Examples of system of one dimensional first order hyperbolic PDEs with source term

I am writing an algorithm to solve a system of one- dimensional first order hyperbolic PDEs with constant coefficient matrices and I require some example problems which are already solved analytically ...
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35 views

Study materials on finite volume method for conservation laws using Python

I am looking for books/lecture notes/open sources to learn coding finite volume/difference methods in Python for hyperbolic conservation laws(kinda beginner's guide to coding)... Any help is ...
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1answer
85 views

2D system of hyperbolic equation (LeVeque)

Again I need help. This time it's exercise 18.1 from LeVeque Finite Volume Methods. Consider the system $q_t+Aq_x+Bq_y=0$ with $A=\begin{pmatrix}3&1\\1&3\end{pmatrix}$ and $B=\begin{pmatrix}0&...
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97 views

$u_t+(u(1-u))_x=a(1-2u)$, transients to steady solution

We consider the non conserving equation $$u_t+(f(u))_x=af'(u)$$ where $a$ is a constant, 0$\leq$ x $\leq$ 1 and $f(u)=u(1-u)$. The steady solution of this equation with boundary condition $u(0)=u_0$ ...
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2answers
177 views

$u_t+(u(1-u))_x=a(1-2u)$, method of characteristics for traffic flow equation with riemann initial data

We consider the non conserving equation $$u_t+(f(u))_x=af'(u)$$ where $a$ is a constant and $f(u)=u(1-u)$. I am trying to solve this equation by method of characteristics with the initial condition $$...
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2answers
117 views

Eigenspaces of 2x2 Jacobian matrix with change of variable

Let $$ A= \begin{pmatrix} v & \rho\\ v\left(v+2\rho\right) & \rho\left(2v+\rho\right) \end{pmatrix} $$ where $\rho>0$, $v=\alpha-\rho$ with $\alpha\in\mathbb{R}$. Show that $A$ has two ...
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1answer
83 views

minmod slope total variation diminishing

This is exercise 6.5 of the book Finite Volume Methods for Hyperbolic Problems by R.J. LeVeque (2002). Show that the minmod slope guarantees that $$ TV(q^n(·, t_n)) ≤ TV(Q^n) \tag{6.23} $$ will be ...
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1answer
76 views

Why is the partial differential equation $u_{t} + a u_{x} = 0$ hyperbolic?

I don't quite understand why this advection equation, with some initial condition $u(0,x) = u_{0}(x)$, is considered hyperbolic (as for instance here). If I apply the test mentioned in Farlow, ...
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1answer
69 views

PDE $u_y+e^uu_x=0$ solution where initial data is discontinuous

Given a pde $$u_y+e^uu_x=0$$ $x \in \mathbb{R}$, $t>0$ with the initial conditions $$u(x,0)=f(x)=\begin{cases} 2 & x<0 \\ 1 & x>0 \end{cases}$$ Solve the pde. My attempt so far, I ...
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2answers
128 views

Check if $u_y+u^2 u_x = 0$ with rectangular initial data has a shock

I have the p.d.e. $$u_y+u^2 u_x =0$$ $x \in \mathbb{R}$, $t>0$ with $$u(x,0)=h(x)= \begin{cases} 0 & x<0 \\ 1 & 0<x<1 \\ 0 & x>1 \end{cases}$$ Check if a shock forms and ...
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1answer
106 views

Stationary shock for Burgers' equation with Murman-Roe scheme

Consider Burgers' equation $u_t + f(u)_x = 0$ where $f(v)=\frac{v^2}{2}$, with initial condition $$u_0(x)=\begin{cases} -1 & x<0 \\ 1& x>0 .\end{cases} $$ It's clear that $u_l=-1<u_r=...
4
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2answers
192 views

Symmetrizability of conservation laws via mathematical entropy

Let $$ \frac{\partial \boldsymbol{u}}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \boldsymbol{f}_j(\boldsymbol{u}) = \boldsymbol{0}, $$ be a system of conservation laws and let us assume ...

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