Discontinuous Galerkin Method for Material Flow Problems

Abstract

For the simulation of material flow problems based on two-dimensional hyperbolic partial differential equations different numerical methods can be applied. Compared to the widely used finite volume schemes we present an alternative approach, namely, the discontinuous Galerkin method, and explain how this method works within this framework. An extended numerical study is carried out comparing the finite volume and the discontinuous Galerkin approach concerning the quality of solutions.

High-Order Numerical Schemes for One-Dimensional Nonlocal Conservation Laws

“…At this regard, it is important to notice that the lack of Riemann solvers for non-local equations limits strongly the choice of the scheme. At the best of our knowledge, two main approaches have been proposed in the literature to treat nonlocal problems: first and second order central schemes like Lax-Friedrichs or Nassyau-Tadmor [2,7,8,17,22] and Discontinuous Galerkin (DG) methods [19]. In particular, the comparative study presented in [19] on a specific model for material flow in two space dimensions, involving density gradient convolutions, encourages the use of DG schemes for their versatility and lower computational cost, but further investigations are needed in this direction.…”

“…At the best of our knowledge, two main approaches have been proposed in the literature to treat nonlocal problems: first and second order central schemes like Lax-Friedrichs or Nassyau-Tadmor [2,7,8,17,22] and Discontinuous Galerkin (DG) methods [19]. In particular, the comparative study presented in [19] on a specific model for material flow in two space dimensions, involving density gradient convolutions, encourages the use of DG schemes for their versatility and lower computational cost, but further investigations are needed in this direction. Besides that, the computational cost induced by the presence of non-local terms, requiring the computation of quadrature formulas at each time step, motivate the development of high order algorithms.…”

Optimal packing of material flow on conveyor belts

“…The literature ranges from the individual computation of part trajectories (microscopic models) to rescaled models describing the evolution of the part density (macroscopic models). Depending on the type of application, numerical methods (Aggarwal et al 2015;Göttlich and Schindler 2015;Evers et al 2015), theoretical investigations (Che et al 2016) and model hierarchies Yu 2002, 2005) are provided. However, optimization issues of such problems are typically considered for fluid problems in general such as Allain et al (2014), McNamara et al (2004), Schlitzer (2000) or Wojtan et al (2006).…”

Existence of a classical solution to complex material flow problems

“…So far, the material flow problem as described earlier has been investigated from a numerical point of view only. Microscopic and macroscopic models have been established and validated against real data , . Analytical considerations are missing until now.…”