Multigrid preconditioners for the hybridised discontinuous Galerkin discretisation of the shallow water equations

17 USC 105 interim-entered record; under review.The article of record as published may be found at https://doi.org/10.1016/j.jcp.2020.109948Numerical climate- and weather-prediction models require the fast solution of the equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have several advantageous properties. They can be used for arbitrary domains and support a structured data layout, which is particularly important on modern chip architectures. For smooth solutions, higher order approximations can be particularly efficient since errors decrease exponentially in the polynomial degree. Due to the wide separation of timescales in atmospheric dynamics, semi-implicit time integrators are highly efficient, since the implicit treatment of fast waves avoids tight constraints on the time step size, and can therefore improve overall efficiency. However, if implicit-explicit (IMEX) integrators are used, a large linear system of equations has to be solved in every time step. A particular problem for DG discretisations of velocity-pressure systems is that the normal Schur-complement reduction to an elliptic system for the pressure is not possible since the numerical fluxes introduce artificial diffusion terms. For the shallow water equations, which form an important model system, hybridised DG methods have been shown to overcome this issue. However, no attention has been paid to the efficient solution of the resulting linear system of equations. In this paper we address this issue and show that the elliptic system for the flux unknowns can be solved efficiently by using a non-nested multigrid algorithm. The method is implemented in the Firedrake library and we demonstrate the excellent performance of the algorithm both for an idealised stationary flow problem in a flat domain and for nonstationary setups in spherical geometry from the well-known testsuite in Williamson et al. (1992) [23]. In the latter case the performance of our bespoke multigrid preconditioner (although itself not highly optimised) is comparable to that of a highly optimised direct solver.EPSRCEPSRCEP/L015684/1UK-Fluids network (EPSRC grant EP/N032861/1

Hybridised multigrid preconditioners for a compatible finite element dynamical core

Compatible finite element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging since the velocity mass matrix is non-diagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate- and weather prediction model LFRic.Comment: 24 pages, 13 figures, 5 tables; accepted for publication in Quarterly Journal of the Royal Meteorological Societ

New discrete and polymeric supramolecular architectures derived from dinuclear Co(II), Ni(II) and Cu(II) complexes of aryl-linked bis-beta-diketonato ligands and nitrogen bases: synthetic, structural and high pressure studies

New examples of nitrogen base adducts of dinuclear Co(II), Ni(II) and Cu(II) complexes of the doubly deprotonated forms of 1,3-aryl linked bis-β-diketones of type [RC([double bond, length as m-dash]O)CH2C([double bond, length as m-dash]O)C6H4C([double bond, length as m-dash]O)CH2C([double bond, length as m-dash]O)R] (L1H2) incorporating the mono- and difunctional amine bases pyridine (Py), 4-ethylpyridine (EtPy), piperidine (pipi), 1,4-piperazine (pip), N-methylmorpholine (mmorph), 1,4-dimethylpiperazine (dmpip) and N,N,N′,N′-tetramethylethylenediamine (tmen) have been synthesised by reaction of the previously reported [Cu2(L1)2]·2.5THF (R = Me), [Cu2(L1)2(THF)2] (R = t-Bu), [Ni2(L1)2(Py)4] (R = t-Bu) and [Co2(L1)2(Py)4] (R = t-Bu) complexes with individual bases of the above type. Comparative X-ray structural studies involving all ten base adduct derivatives have been obtained and reveal a range of interesting discrete and polymeric molecular architectures. The respective products have the following stoichiometries: [Cu2(L1)2(Py)2]·Py (R = Me), [Cu2(L1)2(EtPy)2]·2EtPy (R = t-Bu), [Cu2(L1)2(pipi)2]·2pipi (R = t-Bu), [Cu2(L1)2(mmorph)2] (R = t-Bu), [Cu2(L1)2(tmen)2] (R = t-Bu) and {[Cu2(L1)2(pip)]·pip·2THF}n, [Co2(L1)2(tmen)2] (R = t-Bu), [Ni2(L1)2(Py)4]·dmpip (R = t-Bu), [Ni2(L1)2(pipi)4]·pipi (R = t-Bu) and [Ni2(L1)2(tmen)2] (R = t-Bu). The effect of pressure on the X-ray structure of [Cu2(L1)2(mmorph)2] has been investigated. An increase in pressure from ambient to 9.1 kbar resulted in modest changes to the unit cell parameters as well as a corresponding decrease of 6.7 percent in the unit cell volume. While a small ‘shearing’ motion occurs between adjacent molecular units throughout the lattice, no existing bonds are broken or new bonds formed

Search for CP Violation in the Decay Z -> b (b bar) g

About three million hadronic decays of the Z collected by ALEPH in the years 1991-1994 are used to search for anomalous CP violation beyond the Standard Model in the decay Z -> b \bar{b} g. The study is performed by analyzing angular correlations between the two quarks and the gluon in three-jet events and by measuring the differential two-jet rate. No signal of CP violation is found. For the combinations of anomalous CP violating couplings, ${\hat{h}}_b = {\hat{h}}_{Ab}g_{Vb}-{\hat{h}}_{Vb}g_{Ab}$ and $h^{\ast}_b = \sqrt{\hat{h}_{Vb}^{2}+\hat{h}_{Ab}^{2}}$, limits of \hat{h}_b < 0.59$and$h^{\ast}_{b} < 3.02$ are given at 95\% CL.Comment: 8 pages, 1 postscript figure, uses here.sty, epsfig.st

Production of excited beauty states in Z decays

A data sample of about 3.0 million hadronic Z decays collected by the ALEPH experiment at LEP in the years 1991 through 1994, is used to make an inclusive selection of B~hadron events. In this event sample 4227 \pm 140 \pm 252 B^* mesons in the decay B^* \to B \gamma and 1944 \pm 108 \pm 161 B^{**} mesons decaying into a B~meson and a charged pion are reconstructed. For the well established B^* meson the following quantities areobtained: \Delta M = M_{B^*} - M_{B} = (45.30\pm 0.35\pm 0.87)~\mathrm{MeV}/c^2 and N_{B^*}/(N_B+N_{B^*}) = (77.1 \pm 2.6 \pm 7.0)\%. The angular distribution of the photons in the B^* rest frame is used to measure the relative contribution of longitudinal B^* polarization states to be \sigma_L/(\sigma_L + \sigma_T)= (33 \pm 6 \pm 5)\%. \\ Resonance structure in the M(B\pi)-M(B) mass difference is observed at (424 \pm 4 \pm 10)~\mathrm{MeV}/c^2. Its shape and position is in agreement with the expectation for B^{**}_{u,d} states decaying into B_{u,d}^{(*)} \pi^\pm. The signal is therefore interpreted as arising from them. The relative production rate is determined to be \frac{BR(Z \to b \to B_{u,d}^{**})}{BR(Z \to b \to B_{u,d})} = [27.9 \pm 1.6(stat) \pm 5.9(syst) \phantom{a}^{+3.9}_{-5.6}(model)]\%. where the third error reflects the uncertainty due to different production and decay models for the broad B_{u,d}^{**} states

Tau hadronic branching ratios

From 64492 selected \tau-pair events, produced at the Z^0 resonance, the measurement of the tau decays into hadrons from a global analysis using 1991, 1992 and 1993 ALEPH data is presented. Special emphasis is given to the reconstruction of photons and \pi^0's, and the removal of fake photons. A detailed study of the systematics entering the \pi^0 reconstruction is also given. A complete and consistent set of tau hadronic branching ratios is presented for 18 exclusive modes. Most measurements are more precise than the present world average. The new level of precision reached allows a stringent test of \tau-\mu universality in hadronic decays, g_\tau/g_\mu \ = \ 1.0013 \ \pm \ 0.0095, and the first measurement of the vector and axial-vector contributions to the non-strange hadronic \tau decay width: R_{\tau ,V} \ = \ 1.788 \ \pm \ 0.025 and R_{\tau ,A} \ = \ 1.694 \ \pm \ 0.027. The ratio (R_{\tau ,V} - R_{\tau ,A}) / (R_{\tau ,V} + R_{\tau ,A}), equal to (2.7 \pm 1.3) \ \%, is a measure of the importance of QCD non-perturbative contributions to the hadronic \tau decay widt