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Are upwind schemes such as Godunov type methods superior to central differencing schemes? Do the reasons include superiority in modelling hyperbolic problems with Dirichlet BC's?

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    $\begingroup$ Have you ever tried using a central differencing scheme for a hyperbolic pde? $\endgroup$ – Paul Sep 10 '15 at 21:27
  • $\begingroup$ yes I have, but this is a general question I would like an answer to if possible. $\endgroup$ – melody Sep 11 '15 at 12:50
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To elaborate on Wolfgang's answer: since hyperbolic PDE semi-discretizations with centered differences have purely imaginary eigenvalues, they are only neutrally stable. For linear problems (e.g., the acoustic wave equation or linear Maxwell's equations) this is fine and such methods are commonly used. For instance, in electromagnetics centered differences are referred to simply as "finite difference time domain" (FDTD) methods and are prevalent.

For nonlinear hyperbolic PDEs, this neutral stability generally leads to excitation of nonlinear instabilities. Even if it does not, it still leads to oscillations near points of discontinuity (and discontinuities arise generically in solutions of hyperbolic PDEs). Upwind spatial discretizations shift the eigenvalues -- especially those corresponding to high-frequency modes -- into the left half-plane, leading to more stable (and less oscillatory) solutions.

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Central differencing schemes are not stable if you have advection dominated problems. There really was no other trivial [1] alternative to developing upwind schemes.

[1] There are a few other stabilization methods, of course, but in the finite difference of finite volume context, almost everything that was developed over the first 30 years of numerical computations ended up using upwinding in one variation or another.

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