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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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=...
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 ...