Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

Dark matter coupling to electroweak gauge and Higgs bosons: an effective field theory approach

If dark matter is a new species of particle produced in the early universe as a cold thermal relic (a weakly-interacting massive particle-WIMP), its present abundance, its scattering with matter in direct-detection experiments, its present-day annihilation signature in indirect-detection experiments, and its production and detection at colliders, depend crucially on the WIMP coupling to standard-model (SM) particles. It is usually assumed that the WIMP couples to the SM sector through its interactions with quarks and leptons. In this paper we explore the possibility that the WIMP coupling to the SM sector is via electroweak gauge and Higgs bosons. In the absence of an ultraviolet-complete particle-physics model, we employ effective field theory to describe the WIMP--SM coupling. We consider both scalars and Dirac fermions as possible dark-matter candidates. Starting with an exhaustive list of operators up to dimension 8, we present detailed calculation of dark-matter annihilations to all possible final states, including gamma gamma, gamma Z, gamma h, ZZ, Zh, W+ W-, hh, and f fbar, and demonstrate the correlations among them. We compute the mass scale of the effective field theory necessary to obtain the correct dark-matter mass density, and well as the resulting photon line signals

Tetra­aqua­bis[2-(thio­semicarbazonometh­yl)benzene­sulfonato]calcium(II)

In the title compound, [Ca(C8H8N3O3S2)2(H2O)4], the Ca atom (site symmetry ) adopts a slightly distorted octa­hedral CaO6 geometry and the mol­ecular conformation is stabilized by intra­molecular N—H⋯N inter­actions. In the crystal, the mol­ecules are linked by O—H⋯O, O—H⋯S, N—H⋯O and N—H⋯S hydrogen bonds

Hexaaqua­magnesium bis­(4-amino-3-methyl­benzene­sulfonate)

In the title mol­ecular salt, [Mg(H2O)6](C7H8NO3S)2, the Mg2+ cation lies on an inversion centre. In the crystal, the components are linked by N—H⋯O and O—H⋯O hydrogen bonds, thereby generating sheets parallel to (001)