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I want to solve a set of hyperbolic equations (not the Euler equations) using an upwind type method. I am interested in using a first order upwind scheme and one that is not based on the method of characteristic and does not require the eigenvalues and eigenvector. So far, I have understood that I first need to solve the Reimann problem, to find out the wave speeds (If I'm not mistaken) and then use that information to work out the fluxes at the boundaries using Gudnov type methods. I was wondering if someone could give me a better step to step guide of what I need to do, and recommendation on the type of methods to use?

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  • $\begingroup$ Please edit your question with the following information: is the problem linear or nonlinear? If it is nonlinear, do you expect shocks to appear in the solution? If so, do you know the physically appropriate jump conditions? $\endgroup$ – David Ketcheson Sep 27 '15 at 16:30
  • $\begingroup$ Why don't you want to look at the method of characteristics? $\endgroup$ – spektr Sep 28 '15 at 3:27
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If you like using the finite element method you can use the streamline upwind Petrov/Galerkin (SUPG) which is essentially a stabilized Galerkin method.

Similar modifications can also be made to FDM and FVM.

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