Table of Content

20 September 2022, Volume 4 Issue 3

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ORIGINAL PAPER

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Xiaozhou Li

2022, 4(3):  759-782.  doi:10.1007/s42967-021-00144-3

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Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order $ k+1 $ to order $2k+1$ in the $ L^2 $ norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

Lorenzo Botti, Daniele A. Di Pietro

2022, 4(3):  783-822.  doi:10.1007/s42967-021-00142-5

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We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $ L^2 $-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations

Ohannes A. Karakashian, Michael M. Wise

2022, 4(3):  823-854.  doi:10.1007/s42967-021-00143-4

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The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type, coupled through their nonlinear terms. In our previous work [9], we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes. In this sequel, we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in [9]. The key tool employed to effect our analysis is the dispersive reconstruction developed by Karakashian and Makridakis [20] for related discontinuous Galerkin methods. We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Lu Zhang, Daniel Appelö, Thomas Hagstrom

2022, 4(3):  855-879.  doi:10.1007/s42967-021-00149-y

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We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes

Hendrik Ranocha, Gregor J. Gassner

2022, 4(3):  880-903.  doi:10.1007/s42967-021-00148-z

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Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (https://doi.org/10.5281/zenodo.4054366).

A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

Poorvi Shukla, J. J. W. van der Vegt

2022, 4(3):  904-944.  doi:10.1007/s42967-021-00155-0

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A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $ \Delta t $ satisfy $ h\cong C\Delta t $, with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $ L^\infty $ and $ L^2 $-norms is order $ p+1 $ for polynomial orders $ p=1 $ and $ p=3 $ and order p for polynomial order $ p=2 $.

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields

Dinshaw S. Balsara, Roger Käppeli

2022, 4(3):  945-985.  doi:10.1007/s42967-021-00166-x

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This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger's equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PNPM-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times.

Discontinuous Galerkin Method for Macroscopic Traffic Flow Models on Networks

Lukáš Vacek, Václav Kučera

2022, 4(3):  986-1010.  doi:10.1007/s42967-021-00169-8

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In this paper, we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads. On individual roads, we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters. To solve traffic flows on networks, we construct suitable numerical fluxes at junctions based on preferences of the drivers. We prove basic properties of the constructed numerical flux and the resulting scheme and present numerical experiments, including a junction with complicated traffic light patterns with multiple phases. Differences with the approach to numerical fluxes at junctions from Čanić et al. (J Sci Comput 63: 233–255, 2015) are discussed and demonstrated numerically on a simple network.

Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence

Mohammed Homod Hashim, Akil J. Harfash

2022, 4(3):  1011-1056.  doi:10.1007/s42967-021-00124-7

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In this paper, a finite element scheme for the attraction-repulsion chemotaxis model is analyzed. We introduce a regularized problem of the truncated system. Then we obtain some a priori estimates of the regularized functions, independent of the regularization parameter, via deriving a well-defined entropy inequality of the regularized problem. Also, we propose a practical fully discrete finite element approximation of the regularized problem. Next, we use a fixed point theorem to show the existence of the approximate solutions. Moreover, a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived. In addition, the uniqueness of the fully discrete approximations is preformed. Finally, we discuss the convergence to the fully discrete problem.

Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part II: Time Convergence, Error Analysis and Numerical Results

Mohammed Homod Hashim, Akil J. Harfash

2022, 4(3):  1057-1104.  doi:10.1007/s42967-021-00168-9

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In Hashim and Harfash (Appl. Math. Comput. 2021), using a finite element method, the attraction-repulsion chemotaxis model $ (\mathrm {P}) $ in space is discretised; finite differences were used to do the same in time. Furthermore, the existence of a global weak solution to the system $ (\mathrm {P}_{M} ^{\Delta t}) $\end{document}was demonstrated by means of analysis of the convergence of the fully discrete approximate problem $ (\mathrm {P}_{M, \varepsilon } ^{h, \Delta t} ) $. Moreover, the functions $ \{U, Z, V\} $ were proved to represent a global weak solution to the system $ (\mathrm {P}_{M} ^{\Delta t}) $ by means of a passage to the limit $ \varepsilon , h \rightarrow 0 $ of the approximate system. This paper's purpose is to demonstrate that the solutions can be bounded, independent of M. The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model $ (\mathrm {P}) $, that requires passing to the limits, $ \Delta t \rightarrow 0^{+} $ and $ M \rightarrow \infty $. The time step $ \Delta t $ is subsequently linked to the cutoff parameter $ M > 1 $ by positing a demand that $ \Delta t=o(M^{-1}) $, as $ M \rightarrow \infty $, with the result that the cutoff parameter becomes the only parameter in the problem $ (\mathrm {P}_{M} ^{\Delta t}) $. The solutions can be bounded, independent of M, with the use of special energy estimates, as demonstrated herein. Then, these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives. Additionally, compactness arguments were utilised to explore the convergence of the finite element approximate problem. The conclusion was that a weak solution for $ (\mathrm {P}) $ existed. Finally, we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.

A General Fractional Pollution Model for Lakes

Babak Shiri, Dumitru Baleanu

2022, 4(3):  1105-1130.  doi:10.1007/s42967-021-00135-4

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A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.

On Periodic Oscillation and Its Period of a Circadian Rhythm Model

Miao Feng, Chen Zhang

2022, 4(3):  1131-1157.  doi:10.1007/s42967-021-00146-1

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We theoretically study periodic oscillation and its period of a circadian rhythm model of Neurospora and provide the conditions for the existence of such a periodic oscillation by the theory of competitive dynamical systems. To present the exact expression of the unique equilibrium in terms of parameters of system, we divide them into eleven classes for the Hill coefficient $ n=1 $ or $ n=2 $, among seven classes of which nontrivial periodic oscillations exist. Numerical simulations are made among the seven classes and the models with the Hill coefficient $ n=3 $ or $ n=4 $ to reveal the influence of parameter variation on periodic oscillations and their periods. The results show that their periods of the periodic oscillations are approximately 21.5 h, which coincides with the known experiment result observed in constant darkness.

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Chacha Stephen Chacha

2022, 4(3):  1158-1174.  doi:10.1007/s42967-021-00152-3

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In this study, an iterative algorithm is proposed to solve the nonlinear matrix equation \begin{document}$ X+A^{*}{e}^{X}A=I_{n} $\end{document}. Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation. Comparative analysis for the derived condition numbers and the proposed algorithm are presented. The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches. Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.