Analysis of Convection-Diffusion Problems at Various Peclet Numbers Using Finite Volume and Finite Difference Schemes

Convection-diffusion problems arise frequently in many areas of applied sciences and engineering. In this paper, we solve a convection-diffusion problem by central differencing scheme, upwinding differencing scheme (which are special cases of finite volume scheme) and finite difference scheme at various Peclet numbers. It is observed that when central differencing scheme is applied, the solution changes rapidly at high Peclet number because when velocity is large, the flow term becomes large, and the convection term dominates. Similarly, when velocity is low, the diffusion term dominates and the solution diverges, i.e., mathematically the system does not satisfy the criteria of consistency. On applying upwinding differencing scheme, we conclude that the criteria of consistency is satisfied because in this scheme the flow direction is also considered. To support our study, a test example is taken and comparison of the numerical solutions with the analytical solutions is done. Keywords: Finite volume method, Partial differential equation

Resummed next-to-soft corrections to rapidity distribution of Higgs Boson to NNLO+NNLL‾ \textbf{NNLO} + \overline{\textbf{NNLL} }

We present the resumed predictions consisting of both soft-virtual(SV) as well as next-to-SV(NSV) threshold logarithms to all orders in perturbative QCD for the rapidity distribution of Higgs Boson till $\rm NNLO + \overline{NNLL}$ accuracy at LHC. Using our recent formalism\cite{Ajjath:2020lwb}, the resummation is carried out in the double Mellin space by restricting the NSV contributions only from diagonal $gg$ channel. We perform the inverse Mellin ransformation using the minimal prescription procedure and match it with the corresponding fixed order results. We do a detailed analysis of the numerical impact of the resummed result. The K-factor values at different logarithmic accuracy suggest that the prediction for the rapidity distribution converges and becomes more reliable at $\rm NNLO + \overline{NNLL}$ order. We further observed that the inclusion of resumed NSV contribution improves the renormalisation scale uncertainty at every order in perturbation theory. However, the uncertainty due to factorisation scale increases by the addition of resummed SV+NSV predictions to the fixed order rapidity distribution

Rapidity distribution of pseudo-scalar Higgs boson to NNLOA+NNLL‾\rm{\textbf{NNLO}_A+\overline{\textbf {NNLL}}}

We present the differential predictions for the rapidity distribution of pseudo-scalar Higgs boson through gluon fusion at the LHC. These results are obtained taking into account the soft-virtual (SV) as well as the next-to-soft virtual (NSV) resummation effects to next-to-next-to-leading-logarithmic ($\rm{\overline{NNLL}}$) accuracy and matching them to the approximate fixed order next-to-next-to-leading-order ($\rm{NNLO_A}$) computation. We perform the resummation in two dimensional Mellin space using our recent formalism \cite{Ajjath:2020lwb} by limiting ourselves to the contributions only from gluon-gluon ($gg$) initiated channels. The $\rm{NNLO_A}$ rapidity distribution of pseudo-scalar Higgs is obtained by applying a ratio method on the NNLO rapidity distribution of the scalar Higgs boson. We also present the first analytical results of $\rm{N^3LO}$ rapidity distribution of pseudo-scalar Higgs at SV+NSV accuracy. The phenomenological impacts of $\rm{{NNLO}_A+\overline{{NNLL}}}$ predictions for 13 TeV LHC are studied. We observe that, for $m_A$ =125(700) GeV, the SV+NSV resummation at $\rm{ \overline{NNLL}}$ level brings about 14.76\% (11.48\%) corrections to the $\rm{NNLO}_A$ results at the central scale value of $\mu_R=\mu_F=m_A$. Further, we find that the sensitivity to the renormalisation scale gets improved substantially by the inclusion of NSV resummed predictions at $\rm \overline{NNLL}$ accuracy.Comment: 64 pages, 6 figure

Next-to-soft Virtual Resummation for QCD Observables

We present a framework that resums threshold-enhanced logarithms, originating from soft-virtual and next-to-soft virtual (NSV) contributions in colour-singlet productions, to all orders in perturbation theory. The numerical impacts for these resummed predictions are discussed for the inclusive Drell–Yan di-lepton process up to next-to-next-to-leading logarithmic accuracy, restricting to only diagonal partonic channels

Resummed Higgs boson cross section at next-to SV to NNLO+NNLL‾ \rm NNLO + \rm \overline {NNLL}

We present the resummed predictions for inclusive cross section for the production of Higgs boson at next-to-next-to leading logarithmic ($\rm \overline {NNLL}$) accuracy taking into account both soft-virtual ($\rm SV$) and next-to SV ($\rm NSV$) threshold logarithms. We derive the $N$-dependent coefficients and the $N$-independent constants in Mellin-$N$ space for our study. Using the minimal prescription we perform the inverse Mellin transformation and match it with the corresponding fixed order results. We report in detail the numerical impact of $N$-independent part of resummed result and explore the ambiguity involved in exponentiating them. By studying the K factors at different logarithmic accuracy, we find that the perturbative expansion shows better convergence improving the reliability of the prediction at $\rm NNLO + \overline{NNLL}$ accuracy. For instance, the cross-section at $\rm NNLO + \overline{NNLL}$ accuracy reduces by $3.15\%$ as compared to the $\rm NNLO$ result for the central scale $\mu_R = \mu_F = m_H/2$ at 13 TeV LHC. We also observe that the resummed $\rm SV + NSV$ result improves the renormalisation scale uncertainty at every order in perturbation theory. The uncertainty from the renormalisation scale $\mu_R$ ranges between $(+8.85\% ,-10.12\%)$ at $\rm NNLO$ whereas it goes down to $(+6.54\% , - 8.32\%)$ at $\rm NNLO + \overline{NNLL}$ accuracy. However, the factorisation scale uncertainty is worsened by the inclusion of these NSV logarithms hinting the importance of resummation beyond $\rm NSV$ terms. We also present our predictions for $\rm SV + NSV$ resummed result at different collider energies.Comment: 51 pages, 6 Figure

Strategic Deployment of Distributed Generators Considering Feeders’ Failure Rate and Customers’ Load Type

This paper presents an optimal planning schemetoward the design of distributed generation (DG) integrateddistribution network. The Greedy Search based approach aims todetermine the optimal size and location of DG units in order toachieve designated cost curtailment. The cost curtailmentincludes considerations for investment and maintenance cost ofthe DGs, active power loss cost and reliability level cost of thedistribution network. The deployment strategy consists of addingsuitable size of DGs at appropriate site while considering feeders’failure rate and customer load type. Economic factors specificallyinflation rate and interest rate are taken into account for presentworth evaluation. Also, yearly load growth and hourly and dailyvariations of the load are considered while planning.Additionally, power losses, risk level and voltage profile are alsocomputed to attest the efficacy of the proposed approach.Furthermore, computations are done to calculate amounts of thedetriment due to unreal modeling of the feeders’ failure rate andcustomers’ load type. It is proved that the unreal modeling cannotably impact the results of the problem as well as the optimallocations of the DGs