Energy and entropy stable numerical methods with injected boundary conditions

I denne avhandlingen studerer vi de kompressible Navier-Stokes-likningene formulert med både adiabatiske veggrandvilkår og fjernfeltvilkår. Selv om det er ukjent om disse likningene er velformulerte er de av stor interesse, og de er mye brukt innen numerisk fluiddynamikk. Et resultat av Strang (1964) sier at for ikke-lineære problem diskretisert ved hjelp av en differansemetode som er lineærstabil, er denne metoden konvergent for glatte løsninger. Altså finnes det teori vi kan bruke i analysen av Navier-Stokes-likningene. Derfor studerer vi her teori for velformulerte lineære problem, og stabilitet for numeriske metoder. Dette gjøres både for de kompressible Navier-Stokes-likningene, men også for lineære partielle differensiallikninger som modellproblem. Videre utleder vi entropiestimat for de ikke-lineære Navier-Stokes-likningene, et estimat som virker som et kriterium for den svake løsningen vi leter etter; den skal i tillegg til likningene tilfredsstille termodynamikkens andre lov. Hovedfokuset ved dette arbeidet er stabil håndtering av de adiabatiske veggrandvilkårene og fjernfeltvilkår for Navier-Stokes-likningene. Vi beviser at heftelsesvilkåret (eng.: no- slip condition) kan bli implementert eksakt og fremdeles resultere i et entropiestimat når teknikken brukes i kombinasjon med delvissummasjonsoperatorer (SBP-operatorer) som har diagonale normmatriser og randmatriser. Vi introduserer også en ny metodikk for å sette fjernfeltvilkår, og beviser at den fører til et entropistabilt skjema for de kompressible Navier-Stokes-likningene. Teknikken er i tillegg lineært velformulert. Gjennom hele arbeidet bruker vi SBP-operatorer på grunn av deres gode stabilitetsegenskaper. Vi beviser også at en litt endret versjon av SBP-operatoren som tilnærmer den andrederiverte ved hjelp av endelig-volummetoden gitt av Chandrashekar (2016) er (svakt) konsistent, noe som gjør den egnet til å diskretisere de viskøse leddene i Navier-Stokes-likningene på ustrukturerte gitter.The compressible Navier-Stokes equation subject to both adiabatic wall boundary conditions and far-field boundary conditions are studied in this thesis. Although the well- posedness of these equations is generally unknown, they are of wide interest and are extensively used in computational fluid dynamics. A result by Strang (1964) states that if a non-linear problem is discretised using a difference method that is linearly stable, then this method is convergent for smooth solutions. That is, there exists theory we can use in the analysis of the Navier-Stokes equations. Thus, we study linear well-posedness and stability of numerical schemes both in the context of the compressible Navier-Stokes equations, but also linear partial differential equations as model problems. Furthermore, entropy estimates are derived for the fully non-linear Navier-Stokes equations, which pose as an admissibility criterion for the relevant weak solution we seek; it should additionally satisfy the second law of thermodynamics. The main focus of this work is the stable imposition of the adiabatic wall and far-field boundary conditions for the Navier-Stokes equations. In particular, we prove that the no-slip condition can be imposed strongly and still yield an entropy estimate when used in combination with diagonal-norm summation-by-parts (SBP) operators with diagonal boundary operators. Furthermore, we introduce a new methodology for setting far- field boundary conditions, and prove that it leads to an entropy stable scheme for the compressible Navier-Stokes equations. The procedure is additionally linearly well-posed. Throughout, we employ SBP operators due to their remarkable stability properties. We also prove that a slightly modified version of the finite-volume SBP approximation of the second-derivative given by Chandrashekar (2016) is (weakly) consistent, thus making it suitable for discretising the viscous terms of the Navier-Stokes equations on unstructured grids.Doktorgradsavhandlin

On the timing between terrestrial gamma ray flashes, radio atmospherics, and optical lightning emission

On 25 October 2012 the Reuven Ramaty High Energy Solar Spectroscope Imager (RHESSI) and the Tropical Rainfall Measuring Mission (TRMM) satellites passed over a thunderstorm on the coast of Sri Lanka. RHESSI observed a terrestrial gamma ray flash (TGF) originating from this thunderstorm. Optical measurements of the causative lightning stroke were made by the lightning imaging sensor (LIS) on board TRMM. The World Wide Lightning Location Network (WWLLN) detected the very low frequency (VLF) radio emissions from the lightning stroke. The geolocation from WWLLN, which we also assume is the TGF source location, was in the convective core of the cloud. By using new information about both RHESSI and LIS timing accuracy, we find that the peak in the TGF light curve occurs 230 $\mu$s before the WWLLN time. Analysis of the optical signal from LIS shows that within the uncertainties, we cannot conclude which comes first: the gamma emission or the optical emission. We have also applied the new information about the LIS timing on a previously published event by {\O}stgaard et al. (2012). Also for this event we are not able to conclude which signal comes first. More accurate instruments are needed in order to get the exact timing between the TGF and the optical signal

SBP-SAT schemes for hyperbolic problems

Numerical methods for solving partial differential equations is an important field of study, as it helps us to describe many different processes in the world. An important property of a numerical method, is that it should be a stable approximation of the governing differential equation. For numerical approximations that satisfy a summation-by-parts rule, and that are combined with the simultaneous approximation term technique at the boundaries, energy estimates can be derived to prove stability. The Summation-By-Parts Simultaneous Approximation Term (SBP-SAT) technique was first developed in the context of the finite difference method. More recently, it has been shown that other numerical methods, such as the finite volume method, also can be formulated in the SBP framework. The finite volume method is a popular numerical method, as it can be formulated on unstructured grids. However, Svärd et al. ([1]) showed that some approximations of the second derivative are in fact inconsistent on such grids. Consistency is another key feature of a numerical method. The method should be consistent in order for us to know that we are solving the correct equation. In this thesis, we study the extension of the SBP-SAT technique to the finite volume method. We introduce a methodology for implementing a second derivative approximation on general unstructured grids by including a transformation to a computational domain, where accuracy is expected to be recovered. The numerical experiments demonstrate that full accuracy is not obtained when including the transformation. There are still nodes along and near the boundary that are inconsistent. However, numerical experiments indicate that we have convergence. [1]: M. Svärd, J. Gong, and J. Nordström. An accuracy evaluation of unstructured node-centred finite volume methods. Applied Numerical Mathematics, 58:1142–1158, 2007. doi: 10.1016/j.apnum.2007. 05.002.Masteroppgave i anvendt og beregningsorientert matematikkMAB399MAMN-MA

A new population of terrestrial gamma-ray flashes in the RHESSI data

Terrestrial gamma-ray flashes (TGFs) are the most energetic photon phenomenon occurring naturally on Earth. An outstanding question is as follows: Are these flashes just a rare exotic phenomenon or are they an intrinsic part of lightning discharges and therefore occurring more frequently than previously thought? All measurements of TGFs so far have been limited by the dynamic range and sensitivity of spaceborne instruments. In this paper we show that there is a new population of weak TGFs that has not been identified by search algorithms. We use the World Wide Lightning Location Network (WWLLN) to identify lightning that occurred in 2006 and 2012 within the 800 km field of view of Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). By superposing 740,210 100 ms RHESSI data intervals, centered at the time of the WWLLN detected lightning, we identify at least 141 and probably as many as 191 weak TGFs that were not part of the second RHESSI data catalogue. This supports the suggestion that the global TGF production rate is larger than previously reported

Convergence of Chandrashekar’s Second-Derivative Finite-Volume Approximation

We consider a slightly modified local finite-volume approximation of the Laplacian operator originally proposed by Chandrashekar (Int J Adv Eng Sci Appl Math 8(3):174–193, 2016, https://doi.org/10.1007/s12572-015-0160-z). The goal is to prove consistency and convergence of the approximation on unstructured grids. Consequently, we propose a semi-discrete scheme for the heat equation augmented with Dirichlet, Neumann and Robin boundary conditions. By deriving a priori estimates for the numerical solution, we prove that it converges weakly, and subsequently strongly, to a weak solution of the original problem. A numerical simulation demonstrates that the scheme converges with a second-order rate.publishedVersio

Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.publishedVersio

Radio emissions from double RHESSI TGFs

A detailed analysis of Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) terrestrial gamma ray flashes (TGFs) is performed in association with World Wide Lightning Location Network (WWLLN) sources and very low frequency (VLF) sferics recorded at Duke University. RHESSI clock offset is evaluated and found to experience changes on the 5 August 2005 and 21 October 2013, based on the analysis of TGF-WWLLN matches. The clock offsets were found for all three periods of observations with standard deviations less than 100 {\mu}s. This result opens the possibility for the precise comparative analyses of RHESSI TGFs with the other types of data (WWLLN, radio measurements, etc.) In case of multiple-peak TGFs, WWLLN detections are observed to be simultaneous with the last TGF peak for all 16 cases of multipeak RHESSI TGFs simultaneous with WWLLN sources. VLF magnetic field sferics were recorded for two of these 16 events at Duke University. These radio measurements also attribute VLF sferics to the second peak of the double TGFs, exhibiting no detectable radio emission during the first TGF peak. Possible scenarios explaining these observations are proposed. Double (multipeak) TGFs could help to distinguish between the VLF radio emission radiated by the recoil currents in the +IC leader channel and the VLF emission from the TGF producing electrons

Relativistic electrons from sparks in the laboratory

Discharge experiments were carried out at the Eindhoven University of Technology in 2013. The experimental setup was designed to search for electrons produced in meter-scale sparks using a 1 MV Marx generator. Negative voltage was applied to the high voltage (HV) electrode. Five thin (1 mm) plastic detectors (5 $\rm cm^2$ each) were distributed in various configurations close to the spark gap. Earlier studies have shown (for HV negative) that X-rays are produced when a cloud of streamers is developed 30-60 cm from the negative electrode. This indicates that the electrons producing the X-rays are also accelerated at this location, that could be in the strong electric field from counterstreamers of opposite polarity. Comparing our measurements with modeling results, we find that $\sim$300 keV electrons produced about 30-60 cm from the negative electrode are the most likely source of our measurements. A statistical analysis of expected detection of photon bursts by these fiber detectors indicates that only 20%-45% of the detected bursts could be from soft ($\sim$10 keV) photons, which further supports that the majority of detected bursts are produced by relativistic electrons

Reforming the Teaching and Learning of Foundational Mathematics Courses: An Investigation into the Status Quo of Teaching, Feedback Delivery, and Assessment in a First-Year Calculus Course

publishedVersio

Constraints to do realistic modeling of the electric field ahead of the tip of a lightning leader

Several computer models exist to explain the observation of terrestrial gamma-ray flashes (TGFs). Some of these models estimate the electric field ahead of lightning leaders and its effects on electron acceleration and multiplication. In this paper, we derive a new set of constraints to do more realistic modeling. We determine initial conditions based on in situ measurements of electric field and vertical separation between the main charge layers of thunderclouds. A maximum electric field strength of 50 kV/cm at sea level is introduced as the upper constraint for the leader electric field. The threshold for electron avalanches to develop of 2.86 kV/cm at sea level is introduced as the lower value. With these constraints, we determine a region where acceleration and multiplication of electrons occur. The maximum potential difference in this region is found to be $\sim$52 MV, and the corresponding number of avalanche multiplication lengths is $\sim$3.5. We then quantify the effect of the ambient electric field compared to the leader field at the upper altitude of the negative tip. Finally, we argue that only leaders with the highest potential difference between its tips ($\sim$600 MV) can be candidates for the production of TGFs. However, with the assumptions we have used, these cannot explain the observed maximum energies of at least 40 MeV. Open questions with regard to the temporal development of the streamer zone and its effect on the shape of the electric field remain