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# Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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### How can I check mass conservation when solving the advection equation using an upwind scheme?

My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background Consider ...
1 vote
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### Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?

I am looking for possible numerical methods to solve the PDE $$u_t+c u_x= \frac{-c}{x}u$$ I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
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### For a hyperbolic PDE, is there any proof that the BDF2 method is stable for integrating them?

I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs. Say I have a hyperbolic equation as $\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$. This system is ...
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### Cauchy problem ill-posed?

Find the solution to the Cauchy problem consisting of the wave equation : $$u_{xx}-u_{yy}=0$$ together with initial conditions: $$u(x,0)=0,$$ $$u_{y}(x,0)=g(x)$$ for some known initial datum $g$. Is ...
195 views

### discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
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### Numerical methods that can be written in flux conservative form

I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
160 views

### Shallow water equations (SWE): well-posed initial data for single travelling pulse

This question concerns the 1-dimensional (i.e. only one spatial dimension) shallow water equations (SWE) shown below and how to find initial conditions such that we obtain a travelling pulse/wave ...
219 views

This is my attempt to find the approximate solution of the folowing transport equation \left\{\begin{array}{ll} \partial_{t} u+\partial_{x} u= (x^2-x)t+x^3/3-x^2/2, & t \in(0,0.4), x \in(0,1) \\ ... • 145 1 vote 2 answers 58 views ### What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation? For a scalar quantity u = u(x, t), I'm considering the transport equation \begin{align} u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]} \\ u(0, t) &= u_{\text{in}}, \\ u(x, 0) &= f(x). ... • 133 7 votes 2 answers 438 views ### Numerical flux and source term in FVM (Burger's like equation) I'm trying to solve the following equation with FVMu_t + f(u)_x = g(u) where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
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I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...