Abstract: We present and analyze reliable numerical techniques
for simulating complex flow and transport phenomena related to
natural gas transportation in pipelines. Such kind of problems
are of high interest in the field of petroleum and environmental
engineering. Modeling and understanding natural gas flow and
transformation processes during transportation is important for the
sake of physical realism and the design and operation of pipeline
systems. In our approach a two fluid flow model based on a system
of coupled hyperbolic conservation laws is considered for describing
natural gas flow undergoing hydratization. The accurate numerical
approximation of two-phase gas flow remains subject of strong
interest in the scientific community. Such hyperbolic problems are
characterized by solutions with steep gradients or discontinuities, and
their approximation by standard finite element techniques typically
gives rise to spurious oscillations and numerical artefacts. Recently,
stabilized and discontinuous Galerkin finite element techniques
have attracted researchers’ interest. They are highly adapted to the
hyperbolic nature of our two-phase flow model. In the presentation
a streamline upwind Petrov-Galerkin approach and a discontinuous
Galerkin finite element method for the numerical approximation of
our flow model of two coupled systems of Euler equations are
presented. Then the efficiency and reliability of stabilized continuous
and discontinous finite element methods for the approximation is
carefully analyzed and the potential of the either classes of numerical
schemes is investigated. In particular, standard benchmark problems
of two-phase flow like the shock tube problem are used for the
comparative numerical study.
Abstract: This paper deals with a high-order accurate Runge
Kutta Discontinuous Galerkin (RKDG) method for the numerical
solution of the wave equation, which is one of the simple case of a
linear hyperbolic partial differential equation. Nodal DG method is
used for a finite element space discretization in 'x' by discontinuous
approximations. This method combines mainly two key ideas which
are based on the finite volume and finite element methods. The
physics of wave propagation being accounted for by means of
Riemann problems and accuracy is obtained by means of high-order
polynomial approximations within the elements. High order accurate
Low Storage Explicit Runge Kutta (LSERK) method is used for
temporal discretization in 't' that allows the method to be nonlinearly
stable regardless of its accuracy. The resulting RKDG
methods are stable and high-order accurate. The L1 ,L2 and L∞ error
norm analysis shows that the scheme is highly accurate and effective.
Hence, the method is well suited to achieve high order accurate
solution for the scalar wave equation and other hyperbolic equations.
Abstract: This paper is concerned with the numerical minimization
of energy functionals in BV (
) (the space of bounded variation
functions) involving total variation for gray-scale 1-dimensional inpainting
problem. Applications are shown by finite element method
and discontinuous Galerkin method for total variation minimization.
We include the numerical examples which show the different recovery
image by these two methods.
Abstract: A water surface slope limiting scheme is tested and
compared with the water depth slope limiter for the solution of one
dimensional shallow water equations with bottom slope source term.
Numerical schemes based on the total variation diminishing Runge-
Kutta discontinuous Galerkin finite element method with slope
limiter schemes based on water surface slope and water depth are
used to solve one-dimensional shallow water equations. For each
slope limiter, three different Riemann solvers based on HLL, LF, and
Roe flux functions are used. The proposed water surface based slope
limiter scheme is easy to implement and shows better conservation
property compared to the slope limiter based on water depth. Of the
three flux functions, the Roe approximation provides the best results
while the LF function proves to be least suitable when used with
either slope limiter scheme.