Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability coefficient. The search for high order strong stability time-stepping methods with high order and large allowable time-step had been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we aim to find very high linear order implicit SSP Runge-Kutta methods that are optimal in terms of allowable time-step. Next, we formulate an optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs that have high linear order and nonlinear orders p=2,3,4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems

PO and PN in the wind of the oxygen-rich AGB star IK Tau

Phosphorus-bearing compounds have only been studied in the circumstellar environments (CSEs) of the asymptotic giant branch (AGB) star IRC +10216 and the protoplanetary nebula CRL 2688, both C-rich objects, and the O-rich red supergiant VY CMa. The current chemical models cannot reproduce the high abundances of PO and PN derived from observations of VY CMa. No observations have been reported of phosphorus in the CSEs of O-rich AGB stars. We aim to set observational constraints on the phosphorous chemistry in the CSEs of O-rich AGB stars, by focussing on the Mira-type variable star IK Tau. Using the IRAM 30m telescope and the Submillimeter Array (SMA), we observed four rotational transitions of PN (J=2-1,3-2,6-5,7-6) and four of PO (J=5/2-3/2,7/2-5/2,13/2-11/2,15/2-13/2). The IRAM 30m observations were dedicated line observations, while the SMA data come from an unbiased spectral survey in the frequency range 279-355 GHz. We present the first detections of PN and PO in an O-rich AGB star and estimate abundances X(PN/H2) of about 3x10^-7 and X(PO/H2) in the range 0.5-6.0x10^-7. This is several orders of magnitude higher than what is found for the C-rich AGB star IRC +10216. The diameter (<=0.7") of the PN and PO emission distributions measured in the interferometric data corresponds to a maximum radial extent of about 40 stellar radii. The abundances and the spatial occurrence of the molecules are in very good agreement with the results reported for VY CMa. We did not detect PS or PH3 in the survey. We suggest that PN and PO are the main carriers of phosphorus in the gas phase, with abundances possibly up to several 10^-7. The current chemical models cannot account for this, underlining the strong need for updated chemical models that include phosphorous compounds.Comment: Accepted for publication in Astronomy & Astrophysics, 10 pages, 8 figure

Precise Determination of |V{us}| from Lattice Calculations of Pseudoscalar Decay Constants

Combining the ratio of experimental kaon and pion decay widths, Gamma(K to mu antineutrino{mu} (gamma)) / Gamma(pi to mu \antineutrino (gamma)), with a recent lattice gauge theory calculation of f{K}/f{pi} provides a precise value for the CKM quark mixing matrix element |V{us}|=0.2236(30) or if 3 generation unitarity is assumed |V{us}|=0.2238(30). Comparison with other determinations of that fundamental parameter, implications, and an outlook for future improvements are given

Pressure and non-linear susceptibilities in QCD at finite chemical potentials

When the free energy density of QCD is expanded in a series in the chemical potential, mu, the Taylor coefficients are the non-linear quark number susceptibilities. We show that these depend on the prescription for putting chemical potential on the lattice, making all extrapolations in chemical potential prescription dependent at finite lattice spacing. To put bounds on the prescription dependence, we investigate the magnitude of the non-linear susceptibilities over a range of temperature, T, in QCD with two degenerate flavours of light dynamical quarks at lattice spacing 1/4T. The prescription dependence is removed in quenched QCD through a continuum extrapolation, and the dependence of the pressure, P, on mu is obtained.Comment: 15 pages, 2 figures. Data on chi_uuuu added, discussion enhance

Wigner formula of rotation matrices and quantum walks

Quantization of a random-walk model is performed by giving a qudit (a multi-component wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's $(2j+1)$-dimensional unitary representations of rotations with half-integers $j$'s are useful to analyze the probability laws of quantum walks. For any value of half-integer $j$, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if $(2j+1)$ is even, the probability measure of limit distribution is given by a superposition of $(2j+1)/2$ terms of scaled Konno's density functions, and if $(2j+1)$ is odd, it is a superposition of $j$ terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.Comment: v2: REVTeX4, 15 pages, 4 figure

Continuous-time quantum walk on integer lattices and homogeneous trees

This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page