Computational Science Stack Exchange
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Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

Hyperbolic partial differential equations describe time-dependent wave behavior. Examples of applications modeled by hyperbolic PDEs include:

  • Fluid dynamics
  • Water waves
  • Acoustic waves
  • Electromagnetic waves

Nonlinear hyperbolic PDEs pose a unique computational challenge because of the tendency of discontinuities (shocks) to appear even in solutions with smooth initial data. Numerical methods deal with these discontinuities by using limiters (to avoid oscillation in the computation of derivatives) and Riemann solvers (local solutions based on piecewise constant discontinuous data).

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Which time-integration methods should we use for hyperbolic PDEs?

If we employ the Method of Lines for discretization (separate time and space discretization) of hyperbolic PDEs we obtain after spatial discretization by our favorite numerical method (fx. Finite Volume Method) does it matter in practice which ODE…
Allan P. Engsig-Karup
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How does positivity preservation fit into the implication chain from monotone to monotonicity preserving?

I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws: monotone $\Rightarrow$ $L^1$-contractive $\Rightarrow$ TVD $\Rightarrow$ monotonicity…
Anke
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amplification factor for the Crank Nicolson scheme for the advection equation

I will try one more time being more detailed and careful. Consider the transport equation of the form $$u_t+au_x=0, t\in[0,T],x\in \mathbb{R}, a>0$$ and initial condition $u(0,x)=u_0(x)$. I would like to establish stability of Crank-Nicolson scheme…
Kamil
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Solving compressible inviscid Euler equations with shockwaves in polar coordinates

For the past several weeks I was attempting to adapt Lax-Wendroff or some similar scheme for polar coordinates. The process was complicated due to me being unable to find step-by-step derivations of used schemes, so i had to guess the process for…
Dantragof
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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 instead of the typical water droplet scenario that…
SimpleProgrammer
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public solvers for the time-dependent Schrödinger equation?

Are there efficient public solvers for the time-dependent Schrödinger equation with time-independent Hamiltonian and 2 or 3 degrees of freedom?
Arnold Neumaier
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complexity of flux limiter techniques

My question is not related to any particular problem, rather, I am looking at the equations of the form $$u_t+c(t,x)u_x=0$$ and attempt to solve it numerically. According to http://en.wikipedia.org/wiki/Flux_limiter I can apply Flux limiter method,…
Kamil
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amplification factor of some schemes for the transport equation

Any source of FDM schemes for the transport equation starts by explaining that explicit central differences for the equation of the type $u_t+au_x=0$ can cause oscillations and unconditionally unstable. However, I am questioning about stability of…
Kamil
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hyperbolic equation and characteristics

I am a little confused about the connection between variables for the plain advection pde: $$u_t+au_x=0$$ So initially I thought $x$ and $t$ are independent and $u$ is a function of those, but then we can write the same PDE and say that there are…
Kamil
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