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Generally the step from compressible Euler equations to the Navier-Stokes equations is not that hard, at least the coding part. If you want to implement it with an explicit scheme you have to consider the severe time step restriction of the parabolic contributions. One tricky part, at least for a consistent FV implementation, is the calculation of the ...

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It seems hard to write a general enough FD library thas has wide applicability, since FD methods are not as easy to write for general domains, unlike FEM which uses unstructured grids, for which there is a standard approach. I know of two libraries which might be useful for you Overture: An Object-Oriented Toolkit for Solving Partial Differential Equations ...

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The notation $$\frac{\partial U}{\partial \eta}$$ means usually $$\eta \cdot \nabla U$$. This is correct even if the domain is the interval $[a,b]$. The normal vector on the interval $[a,b]$ @a is $\eta=-1$ and @b $\eta= 1$ both pointing outwards of the domain. Hence in 1D $\frac{\partial U}{\partial \eta}$ means $$\eta\cdot\nabla U=\eta \frac{dU}{dx}$$.

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A general way would be to include the boundary nodes in the definition of $A$ (which will give you a matrix with more columns than rows) and derive $b$ as the contribution of the Dirichlet nodes. This way, other linear terms like convection are readily included. Assume that the matrix $A$ looks like $$A = [A_I | A_\Gamma ]$$ where $A_I$ is the (square) ...

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