2,343 research outputs found
Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows
In this work, we consider the discretization of nonlinear hyperbolic systems
in nonconservative form with the high-order discontinuous Galerkin spectral
element method (DGSEM) based on collocation of quadrature and interpolation
points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp.136--155; Carpenter
et al., SIAM J. Sci. Comput., 36 (2014), pp.~B835-B867). We present a general
framework for the design of such schemes that satisfy a semi-discrete entropy
inequality for a given convex entropy function at any approximation order. The
framework is closely related to the one introduced for conservation laws by
Chen and Shu (J. Comput. Phys., 345 (2017), pp.~427--461) and relies on the
modification of the integral over discretization elements where we replace the
physical fluxes by entropy conservative numerical fluxes from Castro et al.
(SIAM J. Numer. Anal., 51 (2013), pp.~1371--1391), while entropy stable
numerical fluxes are used at element interfaces. Time discretization is
performed with strong-stability preserving Runge-Kutta schemes. We use this
framework for the discretization of two systems in one space-dimension: a
system with a nonconservative product associated to a
linearly-degenerate field for which the DGSEM fails to capture the physically
relevant solution, and the isentropic Baer-Nunziato model. For the latter, we
derive conditions on the numerical parameters of the discrete scheme to further
keep positivity of the partial densities and a maximum principle on the void
fractions. Numerical experiments support the conclusions of the present
analysis and highlight stability and robustness of the present schemes
Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
In this paper we generalize to non-uniform grids of quad-tree type the
Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput.,
2001), thus obtaining a truly two-dimensional non-oscillatory third order
reconstruction with a very compact stencil and that does not involve
mesh-dependent coefficients. This latter characteristic is quite valuable for
its use in h-adaptive numerical schemes, since in such schemes the coefficients
that depend on the disposition and sizes of the neighboring cells (and that are
present in many existing WENO-like reconstructions) would need to be recomputed
after every mesh adaption.
In the second part of the paper we propose a third order h-adaptive scheme
with the above-mentioned reconstruction, an explicit third order TVD
Runge-Kutta scheme and the entropy production error indicator proposed by Puppo
and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on
the choice of the parameters controlling the mesh adaption, we demonstrate with
many numerical tests that the scheme can compute numerical solution whose error
decays as , where is the average
number of cells used during the computation, even in the presence of shock
waves, by making a very effective use of h-adaptivity and the proposed third
order reconstruction.Comment: many updates to text and figure
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