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.
Questions tagged [hyperbolic-equations]
478 questions
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votes
0 answers
Is there a physical interpretation of the "test function" for weak solutions to differential equations?
Background
Consider the inviscid Burgers' equation.
\begin{align*}
u_t + f(u)_x&=0
\end{align*}
or, using $f(u)=\frac{1}{2}u^2$,
\begin{align*}
u_t + u u_x &=0
\end{align*}
In the case of non-smooth initial data $u_0(x)$, a classical solution does…
nwsteg
- 329
2
votes
1 answer
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 $u,v$ are two solutions with the intial data satisfying…
Anju George
- 471
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votes
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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 differential operator. That means for…
Martin
- 31
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votes
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Hyperbolic conservation laws with integral source term
Suppose
$u_t+\alpha u_x =\int\limits_{b-h}^{b} u(t,x) dx $
$u=u(x,t), (x,t)\in ([a,b]\times [0,T])$
with initial data $ u(x,0)=u_0(x)$, h is fixed xonstant.
I want to solve this equation by using finite volume method (FVM). I can apply FVM for …
Sandy
- 41
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vote
0 answers
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 intersect and at that point, you calculate the value. My…
Upstart
- 2,613
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What is the use of the notion of consistency for Riemann solvers?
Consistency of a Riemann solver appear in the theory of conservation laws and systems of conservation laws. It is defined as follow:
A Riemann solver $\mathcal{R}$ is consistent if it satisfies the following two properties:
Fix $u_l ,u_r$
For any…
incas
- 805
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votes
1 answer
integral inequality in energy inequality
I encountered these two inequalities in reading Sogge's lectures on nonlinear wave equations (page 17). It seems natural and straightforward such that the author didn't give any hint but I cannot work out a prrof. Appreciate any hints.
(1) Let…
Bowen Zhao
- 97
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votes
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1D advection with space-dependent velocity
I'm trying to follow http://www2.mpia-hd.mpg.de/~dullemon/lectures/fluiddynamics08/chap_2_hyperbolic.pdf and am confused by equation 2.5 and 2.6. To summarize, they define
$\xi = \int^{x}_{x_0} \frac{dx'}{u(x')}$ to state the solution to the…
