Istituto di Matematica Applicata e Tecnologie Informatiche     
Cornetti G., Manzini G. An adjoint discontinuous Galerkin method for conservation laws. Preprint ercim.cnr.ian//1997-1049, 1997.
A two-points numerical flux is presented in the framework of the discontinuous Galerkin method for scalar conservation laws and systems. In the scalar case, the numerical flux is proved to be Lipschitz continuous and monotone for a class of conservation laws. This allows to state the convergence to the unique entropy solution of the method with piecewise constant interpolation. The numerical flux is related to the adjoint problem: following this idea the method is extended to systems of conservation laws. No approximate Riemann solver is needed and the matrix which has to be diagonalized has a simpler structure than the Jacobian matrix. Numerical examples validate the theoretical results and show the convergence of the scheme when applied to systems of hyperbolic equations arising in fluid dynamics, such as the Shallow Water and the compressible Euler equations.
Subject Discontinuous Galerkin method, Finite element method, Shock capturing, Monotone flux

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