For the simulation of material flow problems based on twodimensional 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.
The modeling and simulation of material flow problems is motivated by engineering plant planning [
Macroscopic models, or conservation laws, are used in different engineering areas, for example, traffic flow [
In this work, we are concerned with a twodimensional nonlocal hyperbolic conservation law that has been originally introduced [
The paper is structured as follows. A short introduction of the macroscopic model taken from [
We consider a conveyor belt with the given geometry presented in Figure
(a) Geometry of the conveyor belt. (b) Static velocity field used for numerical simulations.
We briefly recall the conservation law for the evolution of the part density
Equation (
In contrast, the dynamic component
The nonlocal operator
The boundary conditions of (
Now we present suitable numerical methods to solve the conservation law (
For both simulation approaches, we assume that the discontinuous flux function in (
The following procedure is based on a finite volume method with dimensional splitting; see [
Summarizing, we have to solve the coupled scheme
We now present an alternative approach to solve (
In the following derivation, we assume that the flux function
We consider a finite element discretization of the spatial domain
A finite element discretization (triangulation) of a domain
Let
Nodal points of the basis for linear, quadratic, and cubic triangle elements
An approximation of the solution (
Interface of two neighboring triangles
The coefficient matrices
It is well known that nonlinear conservation laws might lead to shocks or discontinuities in solutions. However, the polynomial approximation of solutions of the DG method is not able to prescribe discontinuities so far. If we apply the previous DG method to problems with shock solutions, the following problems will occur:
The appearance of artificial and persistent oscillations around the point of discontinuity.
The loss of pointwise convergence at the point of discontinuity.
Note that a high order polynomial basis on the elements gives a high order accuracy of the scheme for smooth solutions. However, the DG method handles discontinuities with persistent oscillations that distort the approximate solution or influence the stability properties. Therefore, we propose the following filter approach in stabilizing the computations and in reducing the oscillations.
The filter approach [
In the following, we consider the canonical basis
Examples of how the filter function (
Since filtering usage should be used both as minimal as possible and as much as needed, this is necessary to stabilize the method, reduce oscillatory solutions, and reduce artificial viscosity.
In particular, the dispersive term
Note that the weights
The DG approximation leads to a system of
As a result, the DG computation procedure is illustrated by the following steps:
Computation of
Reconstruction of the updated solution
where
Finally, we present computational results comparing the finite volume approach and the discontinuous Galerkin method presented in Section
All computations are performed on the same platform, namely, a 3.0 GHz Dual Core computer with 8 GB RAM, and all algorithms are implemented in MATLAB.
First, we compare the quality of the two methods to numerically solve (
The field
The grid sizes of the finite volume approach with dimensional splitting are chosen as
The discontinuous Galerkin method uses a triangulation
The results are shown in Figure
Results of the finite volume method with splitting:
Time
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Time
Time
Time
Time
Time
Time
In all plots, we recognize a weak dispersing of density (cf. Figures
The following question rises: what mesh grid sizes and what polynomial degrees are necessary to ensure good approximations due to the discontinuous Galerkin method. In the following, the previous example is computed again by the DG method with different triangulations and polynomial degrees. We test our problem on 3 different mesh grid sizes
Results of the discontinuous Galerkin method with different triangulations (
However, a rough triangulation or a low polynomial degree causes bad approximations (cf. Figure
The computation times of the DG method with respect to the meshsizes and polynomial degrees are shown in Tables
Computation times of the discontinuous Galerkin method (simulation process) with different grid sizes





1  7.14 s  12.30 s  13.42 s 
3  9.30 s  18.63 s  51.70 s 
5  17.31 s  46.29 s  — 
7  30.10 s  111.94 s  — 
9  49.58 s  —  — 
11  88.70 s  —  — 
Computation times in seconds for the convolution preprocessing due to the grid size





1  0.06 s  1.12 s  1.73 s 
3  0.48 s  3.88 s  79.99 s 
5  2.01 s  60.82 s  — 
7  5.44 s  420.08 s  — 
9  47.11 s  —  — 
11  183.86 s  —  — 
The computing time required for the calculation of the finite volume approach is about 788.21 s. Consequently, the DG method is quite faster than the finite volume approach for all presented settings. However, the computing times and the memory requirements of the DG preprocessing increase enormously since the computation of the convolution in one nodal point requires at most
Let us summarize. The discontinuous Galerkin method is able to approximate accurately the extended flow equations on complex geometric domains. However, the presented example consists only of a rectangleshaped domain and it is not necessary to use methods for complex geometries (cf. regular grids). As already seen, the DG method needs a very time and memory consuming preprocessing due to the convolution. Hence, it is very expensive to apply small step sizes
The macroscopic model (
Pedestrian models as [
We solve the simplified equation (
The results of the finite volume approach are shown in Figure
Numerical solution of the simplified model (
Time
Time
Time
Time
Time
Time
Time
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Time
Figure
Comparison of the results of the discontinuous Galerkin method with different meshsizes
We have presented a novel numerical simulation algorithm, the discontinuous Galerkin method, to compute the movement of material flow on conveyor belts. The numerical difficulties arise from the predefined geometry of the setting and the flux function consisting of a nonlocal term including a convolution. We have tested the performance of the discontinuous Galerkin method against a finite volume scheme and observed satisfactory results. In addition to the good qualitative behavior of the numerical results, we also detected and verified solution artifacts as lane formation in both numerical approaches.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was financially supported by the German Research Foundation (DFG), DFG grant OptiFlow (ProjectID GO 1920/31). Special thanks go to Veronika Schleper for fruitful and inspiring discussions.