On leapfrog-Chebyshev schemes for second-order differential equations

In this thesis the efficient time integration of semilinear second-order ordinary differential equations is investigated. Based on the leapfrog (Störmer, Verlet) scheme a new class of explicit two-step schemes is constructed by utilizing Chebyshev polynomials. For deriving rigorous error bounds of these leapfrog-Chebyshev (LFC) schemes a more general class of two-step schemes is introduced. Precise conditions are stated for this general class guaranteeing stability as well as second-order convergence in time. In addition, the influence of the starting value is analyzed in detail. Furthermore, by combining the leapfrog scheme with this general class of schemes a class of multirate two-step methods is constructed. Sufficient conditions for the stability of these schemes are derived as well as error bounds showing the second-order convergence in time. For both the LFC schemes and the multirate schemes if equipped with the LFC schemes it is shown that in specific situations they outperform the leapfrog scheme. Numerical examples are provided to illustrate the theoretical results

Error analysis of multirate leapfrog-type methods for second-order semilinear ODEs

In this paper we consider the numerical solution of second-order semilinear differential equations, for which the stiffness is induced by only a few components of the linear part. For such problems, the leapfrog scheme suffers from severe restrictions on the step size to ensure stability. We thus propose a general class of multirate leapfrog-type methods which allows to use step sizes which are independent on the stiff part of the equation and also very efficient to implement. This class comprises local time-stepping schemes [5, 7] but also locally implicit or locally trigonometric integrators. Our main contribution is a rigorous error and stability analysis with special emphasis on explicit multirate methods, which are based on stabilized leapfrog-Chebyshev polynomials introduced in [4]

Error analysis of second-order locally implicit and local time-stepping methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong CFL condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time

On leapfrog-Chebyshev schemes

This paper is dedicated to the improvement of the efficiency of the leapfrog method for linear and semilinear second-order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed for linear problems that exhibit a much weaker CFL condition than the leapfrog method (at a higher cost). However, these methods fail to produce the correct long-time behavior of the exact solution which can result in a bad approximation quality. In this paper we introduce a new class of leapfrog-Chebyshev methods for semilinear problems. For the linear part, we use Chebyshev polynomials while the nonlinearity is treated by the standard leapfrog method. The method can be viewed as a multirate scheme because the nonlinearity is evaluated only once in each time step whereas the number of evaluations of the linear part corresponds to the degree of the Chebyshev polynomial. In contrast to existing literature (which is restricted to linear problems), we suggest to stabilize the scheme and we introduce a new starting value required for the two-step method. A new representation formula for the approximations obtained by using generating functions allows us to fully understand the stability and the long-time behavior of the stabilized and the unstabilized scheme. In particular, for linear problems we prove that these new schemes approximately preserve a discrete energy norm over arbitrarily long times. The stability analysis shows that stabilization is essential to guarantee a favorable CFL condition for the multirate scheme, which is closely related to local time-stepping schemes. We also show error bounds of order two for semilinear problems and that a special choice of the stabilization yields order four for linear problems. Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations

Measurement of the top quark forward-backward production asymmetry and the anomalous chromoelectric and chromomagnetic moments in pp collisions at √s = 13 TeV

Abstract The parton-level top quark (t) forward-backward asymmetry and the anomalous chromoelectric (d̂ t) and chromomagnetic (μ̂ t) moments have been measured using LHC pp collisions at a center-of-mass energy of 13 TeV, collected in the CMS detector in a data sample corresponding to an integrated luminosity of 35.9 fb−1. The linearized variable AFB(1) is used to approximate the asymmetry. Candidate t t ¯ events decaying to a muon or electron and jets in final states with low and high Lorentz boosts are selected and reconstructed using a fit of the kinematic distributions of the decay products to those expected for t t ¯ final states. The values found for the parameters are AFB(1)=0.048−0.087+0.095(stat)−0.029+0.020(syst),μ̂t=−0.024−0.009+0.013(stat)−0.011+0.016(syst), and a limit is placed on the magnitude of | d̂ t| < 0.03 at 95% confidence level. [Figure not available: see fulltext.

Measurement of b jet shapes in proton-proton collisions at root s=5.02 TeV

We present the first study of charged-hadron production associated with jets originating from b quarks in proton-proton collisions at a center-of-mass energy of 5.02 TeV. The data sample used in this study was collected with the CMS detector at the CERN LHC and corresponds to an integrated luminosity of 27.4 pb(-1). To characterize the jet substructure, the differential jet shapes, defined as the normalized transverse momentum distribution of charged hadrons as a function of angular distance from the jet axis, are measured for b jets. In addition to the jet shapes, the per-jet yields of charged particles associated with b jets are also quantified, again as a function of the angular distance with respect to the jet axis. Extracted jet shape and particle yield distributions for b jets are compared with results for inclusive jets, as well as with the predictions from the pythia and herwig++ event generators.Peer reviewe

Measurement of the Jet Mass Distribution and Top Quark Mass in Hadronic Decays of Boosted Top Quarks in pp Collisions at root s=13 TeV

A measurement is reported of the jet mass distribution in hadronic decays of boosted top quarks produced in pp collisions at root s = 13 TeV. The data were collected with the CMS detector at the LHC and correspond to an integrated luminosity of 35.9 fb(-1). The measurement is performed in the lepton + jets channel of t (t) over bar events, where the lepton is an electron or muon. The products of the hadronic top quark decay t -> bW -> bq (q) over bar' are reconstructed as a single jet with transverse momentum larger than 400 GeV. The t (t) over bar cross section as a function of the jet mass is unfolded at the particle level and used to extract a value of the top quark mass of 172.6 +/- 2.5 GeV. A novel jet reconstruction technique is used for the first time at the LHC, which improves the precision by a factor of 3 relative to an earlier measurement. This highlights the potential of measurements using boosted top quarks, where the new technique will enable future precision measurements.Peer reviewe

Measurement of the azimuthal anisotropy of Y(1S) and Y(2S) mesons in PbPb collisions at root s(NN)=5.02 TeV

The second-order Fourier coefficients (v(2)) characterizing the azimuthal distributions of Y(1S) and Y(2S) mesons produced in PbPb collisions at root s(NN) = 5.02 TeV are studied. The Y mesons are reconstructed in their dimuon decay channel, as measured by the CMS detector. The collected data set corresponds to an integrated luminosity of 1.7 nb(-1). The scalar product method is used to extract the v2 coefficients of the azimuthal distributions. Results are reported for the rapidity range vertical bar y vertical bar < 2.4, in the transverse momentum interval 0 < pT < 50 GeV/c, and in three centrality ranges of 10-30%, 30-50% and 50-90%. In contrast to the J/psi mesons, the measured v(2) values for the Y mesons are found to be consistent with zero. (C) 2021 The Author(s). Published by Elsevier B.V.Peer reviewe

Search for top squark pair production in a final state with two tau leptons in proton-proton collisions at root s=13 TeV

A search for pair production of the supersymmetric partner of the top quark, the top squark, in proton-proton collision events at s = 13 TeV is presented in a final state containing hadronically decaying tau leptons and large missing transverse momentum. This final state is highly sensitive to high-tan beta or higgsino-like scenarios in which decays of electroweak gauginos to tau leptons are dominant. The search uses a data set corresponding to an integrated luminosity of 77.2 fb(-1), which was recorded with the CMS detector during 2016 and 2017. No significant excess is observed with respect to the background prediction. Exclusion limits at 95% confidence level are presented in the top squark and lightest neutralino mass plane within the framework of simplified models, in which top squark masses up to 1100 GeV are excluded for a nearly massless neutralino.Peer reviewe

Search for charged Higgs bosons decaying into a top and a bottom quark in the all-jet final state of pp collisions at root s=13 TeV

A search for charged Higgs bosons (H-+/-) decaying into a top and a bottom quark in the all-jet final state is presented. The analysis uses LHC proton-proton collision data recorded with the CMS detector in 2016 at root s = 13 TeV, corresponding to an integrated luminosity of 35.9 fb(-1). No significant excess is observed above the expected background. Model-independent upper limits at 95% confidence level are set on the product of the H-+/- production cross section and branching fraction in two scenarios. For production in association with a top quark, limits of 21.3 to 0.007 pb are obtained for H-+/- masses in the range of 0.2 to 3 TeV. Combining this with a search in leptonic final states results in improved limits of 9.25 to 0.005 pb. The complementary s-channel production of an H-+/- is investigated in the mass range of 0.8 to 3 TeV and the corresponding upper limits are 4.5 to 0.023 pb. These results are interpreted using different minimal supersymmetric extensions of the standard model.Peer reviewe