1,329 research outputs found
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation of the two-phase flow system. After
discretizing in time with diagonally implicit Runge-Kutta schemes the resulting
systems of nonlinear algebraic equations are solved with Newton's method and
the arising systems of linear equations are solved efficiently and in parallel
with an algebraic multigrid method. The new scheme is investigated for various
test problems from the literature and is also compared to a cell-centered
finite volume scheme in terms of accuracy and time to solution. We find that
the method is accurate, robust and efficient. In particular no post-processing
of the DG velocity field is necessary in contrast to results reported by
several authors for decoupled schemes. Moreover, the solver scales well in
parallel and three-dimensional problems with up to nearly 100 million degrees
of freedom per time step have been computed on 1000 processors
Local discontinuous Galerkin methods for fractional ordinary differential equations
This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence in the norm and
superconvergence of order at the downwind point of each
element. Here is the degree of the approximation polynomial used in an
element and () represents the order of the one-term
FODEs. A generalization of this includes problems with classic 'th-term
FODEs, yielding superconvergence order at downwind point as
. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure
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