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Flux computation in unstructured grids

Posted July 16, 2010 at 10:49 by praveen (CFD, Numerical Analysis, Scientific Computing)

Consider finite volume scheme on unstructured grids for the Euler equations.

Let n=(n_x,n_y,n_z) be normal to a cell face and whose magnitude is equal to face area. Let Q be the conserved vector. The finite volume update equation using forward Euler time discretization is

Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk})

Here n_{jk} is a normal vector pointing from current cell "j" into the neighbouring cell...
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Adjoint consistency of DG method

Posted July 8, 2010 at 11:54 by praveen (CFD, Numerical Analysis, Scientific Computing)

Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations
SIAM J. Numer. Anal. Volume 45, Issue 6, pp. 2671-2696 (2007)
http://dx.doi.org/10.1137/060665117

This paper gives a notion of consistent adjoint discretizations for DG methods. However the definition seems to be restricted to smooth solutions only. For the case of a conservation law, they show that boundary conditions of the primal problem must be appropriate for the corresponding adjoint discretization...
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Convergence of discrete adjoints

Posted July 6, 2010 at 12:12 by praveen (CFD, Numerical Analysis, Scientific Computing)

Interesting result on discrete adjoints: They show that discrete adjoint for scalar conservation law converges pointwise everywhere except at shocks, provided the numerical dissipation increases as the grid is refined. The number of points across the shock must increase as the grid is refined.

http://link.aip.org/link/?SNA/48/882/1&agg=rss

Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 1: Linearized Approximations...
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