Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

SUPG-YZβ formulation for solving convection-dominated steady linear reaction-convection-diffusion equations

In this computational study, stabilized finite element solutions of convection-dominated steady linear reaction-convection-diffusion equations are examined. Although the standard Galerkin finite element method (GFEM) is one of the most robust, efficient, and reliable methods for many engineering simulations, it suffers from instability issues in solving convection-dominated problems. To this end, this work deals with a stabilized version of the standard GFEM, called the streamline-upwind/Petrov-Galerkin (SUPG) formulation, to overcome the instability issues in solving such problems. The stabilized formulation is further supplemented with YZβ shock-capturing to provide additional stability around sharp gradients. A comprehensive set of test computations is provided to compare the results obtained by using the GFEM, SUPG, and SUPG-YZβ formulations. It is observed that the GFEM solutions involve spurious oscillations for smaller values of the diffusion parameter, as expected. These oscillations are significantly eliminated when the SUPG formulation is employed. It is also seen that the SUPG-YZβ formulation provides better solution profiles near steep gradients, in general.

Evaluating the Parameters of a Systematic Long-Term Measurement of the Concentration and Mobility of Air Ions in the Environment inside Císařská Cave

Determining the concentration and mobility of light air ions is an indispensable task to ensure the successful performance and progress of various operations within multiple fields and branches of human activity. This article discusses a novel methodology for measuring air ions in an environment with high relative humidity, such as that of a cave. Compared to common techniques, the proposed method exhibits a lower standard deviation and analyses the causes of spurious oscillations in the measured patterns obtained from FEM-based numerical simulations on the one hand and a model with concentrated parameters on the other. The designed ion meter utilises a gerdien tube to facilitate long-term measurement in cold and very humid spaces, an operation that can be very problematic if executed with other devices. Importantly, the applied procedure for calculating the mobility spectra of air ions from the acquired saturation characteristics is insensitive to fluctuations and noises in the measured patterns, and it also enables us to confirm the presence of very mobile air ions generated by fragmenting water droplets. During the sensing cycles, the concentration of light negative ions was influenced by the active gerdien tube. For the investigated cave, we had designed a measuring sequence to cover not only the time dependence of the concentration of light negative ions but also their mobility; this approach then allowed monitoring the corresponding impact of the patients’ presence in the cave, an effect neither described nor resolved thus far. Such comprehensive research, especially due to its specific character, has not been frequently conducted or widely discussed in the literature; the efforts characterised herein have therefore expanded the relevant knowledge and methodology, thus contributing towards further advancement in the field.

Optimization of Lumped Parameter Models to Mitigate Numerical Oscillations in the Transient Responses of Short Transmission Lines

The Lumped Parameter Model (LPM) is a known approach to represent overhead transmission lines (TLs), especially when these elements comprehend a few tens of kilometers. LPMs employ a large number of cascaded π-circuits to compute accurately the transient responses. These responses contain numerical spurious oscillations (NSO) characterized by erroneous peaks which distort the transient responses, mainly their peak values. Two modified LPM topologies composed of damping resistances inserted along the longitudinal or transversal branches of the cascaded π-circuits offer significant mitigations in the NSO. In this paper, in an effort to have the maximum mitigation of the NSO and low distortion in the transient responses, two modified topologies with optimized damping resistances are proposed to represent short TLs. Results demonstrate expressive attenuation in the peaks of NSO which reflect good agreement in comparison with the responses computed by the Bergeron’s line model. The mitigation of the NSO is carried out directly in the time domain and it does not require either analog or digital filters.Furthermore, no frequency-to-time transformations are necessary in this procedure. These alternative topologies can be incorporated into any electromagnetic transient program to study switching operations in power systems.

Towards Building the OP-Mapped WENO Schemes: A General Methodology

A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions, but in the meantime prevent spurious oscillations in the solving of hyperbolic conservation laws with long output times. Our goal for this article was to address this widely known problem. In our previous work, the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long simulations, and hence it was indicated that the OP criterion plays a critical role in the maintenance of low-dissipation and robustness for mapped WENO schemes. Thus, in our present work, we firstly defined the family of mapped WENO schemes, whose mappings meet the OP criterion, as OP-Mapped WENO. Next, we attentively took a closer look at the mappings of various existing mapped WENO schemes and devised a general formula for them. That helped us to extend the OP criterion to the design of improved mappings. Then, we created a generalized implementation of obtaining a group of OP-Mapped WENO schemes, named MOP-WENO-X, as they are developed from the existing mapped WENO-X schemes, where the notation “X” is used to identify the version of the existing mapped WENO scheme. Finally, extensive numerical experiments and comparisons with competing schemes were conducted to demonstrate the enhanced performances of the MOP-WENO-X schemes.

Exclusion of the Possibility of “False Ripples” From Ripple Band High-Frequency Oscillations Recorded From Scalp Electroencephalogram in Children With Epilepsy

AimRipple-band epileptic high-frequency oscillations (HFOs) can be recorded by scalp electroencephalography (EEG), and tend to be associated with epileptic spikes. However, there is a concern that the filtration of steep waveforms such as spikes may cause spurious oscillations or “false ripples.” We excluded such possibility from at least some ripples by EEG differentiation, which, in theory, enhances high-frequency signals and does not generate spurious oscillations or ringing.MethodsThe subjects were 50 pediatric patients, and ten consecutive spikes during sleep were selected for each patient. Five hundred spike data segments were initially reviewed by two experienced electroencephalographers using consensus to identify the presence or absence of ripples in the ordinary filtered EEG and an associated spectral blob in time-frequency analysis (Session A). These EEG data were subjected to numerical differentiation (the second derivative was denoted as EEG″). The EEG″ trace of each spike data segment was shown to two other electroencephalographers who judged independently whether there were clear ripple oscillations or uncertain ripple oscillations or an absence of oscillations (Session B).ResultsIn Session A, ripples were identified in 57 spike data segments (Group A-R), but not in the other 443 data segments (Group A-N). In Session B, both reviewers identified clear ripples (strict criterion) in 11 spike data segments, all of which were in Group A-R (p < 0.0001 by Fisher’s exact test). When the extended criterion that included clear and/or uncertain ripples was used in Session B, both reviewers identified 25 spike data segments that fulfilled the criterion: 24 of these were in Group A-R (p < 0.0001).DiscussionWe have demonstrated that real ripples over scalp spikes exist in a certain proportion of patients. Ripples that were visualized consistently using both ordinary filters and the EEG″ method should be true, but failure to clarify ripples using the EEG″ method does not mean that true ripples are absent.ConclusionThe numerical differentiation of EEG data provides convincing evidence that HFOs were detected in terms of the presence of such unusually fast oscillations over the scalp and the importance of this electrophysiological phenomenon.

Re-evaluating rhythmic attentional switching: Spurious oscillations from shuffling-in-time

How does attention help to focus perceptual processing on the important parts of a visual scene? Although the neural and perceptual effects of attention were traditionally assumed to be sustained over time, the field is converging on a dramatically different view: that covert attention rhythmically switches between objects at 3-8 Hz. Here I demonstrate that ubiquitous analyses in this literature conflate rhythmic oscillations with aperiodic temporal structure. Using computational simulations, I show that the behavioral oscillations reported in this literature could reflect aperiodic dynamics in attention, rather than periodic rhythms. I then propose two analyses (one novel and one widely used in climate science) that discriminate between periodic and aperiodic structure in behavioral time-series. Finally, I apply these alternative analyses to published data-sets, and find no evidence for rhythms in attentional switching after accounting for aperiodic temporal structure. Attention shows rich temporal structure. The techniques presented here will help to clarify the periodic and aperiodic dynamics of perception and cognition.

Unequal order T-spline meshes for fracture in poroelastic media

Spline-based meshes allow for a higher inter-element continuity. For coupled problems, e.g. poroelasticity, different meshes with different orders of interpolation are normally used for the various fields in order to avoid spurious oscillations. When including discontinuities in these meshes, there exist several options for the discretisation. Herein we will discuss two options which use T-splines, one aiming at a minimum number of degrees of freedom around the crack tip, the other trying to maximise this number. Both meshes retain a higher-order continuity along the fracture, but the mesh which maximises the number of degrees of freedom mesh introduces two additional degrees of freedom around the crack tip to allow for a sharper crack. The two discretisations are used to simulate a pressurised fracture inside a poroelastic material and the results are compared to results obtained using a Non-Uniform Rational B-Spline (NURBS) mesh. A comparison between the two discretisations shows the effect of including additional degrees of freedom close to the crack tip. However, both meshes yield similar results further away from the crack tip. It is shown that both T-spline meshes capture a fully closed discontinuity at the fracture tip, whereas the NURBS mesh retains a small opening due to the discontinuity which exists for the cracked as well as the intact elements. A fully closed fracture aperture results in T-splines with a lower discontinuity pressure compared to NURBS, making T-splines more suitable for simulations in which the fracture propagation is limited by the fluid transport within the fracture.