Integrating Singular Functions on the Sphere

We obtain rigorous results concerning the evaluation of integrals on the two sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure.Comment: 15 pages, revte

Numerical Treatment of Interfaces for Second-Order Wave Equations

In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. We show that it is possible to implement an interface scheme of “penalty” type for the second-order wave equation, similar to the ones used for first-order hyperbolic and parabolic equations, and the second-order scheme used by Mattsson et al. 2008. These schemes, known as SAT schemes for finite difference approximations and penalties for spectral ones, and ours share similar properties but in our case one needs to pass at the interface a smaller amount of data than previously known schemes. This is important for multi-block parallelizations in several dimensions, for it implies that one obtains the same solution quality while sharing among different computational grids only a fraction of the data one would need for a comparable (in accuracy) SAT or Mattsson et al.’s scheme. The semi-discrete approximation used here preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge–Kutta method. This is crucial, since the explicit Runge–Kutta method would be impractical given the severe restrictions that arise from the stiff parts of the equations.Fil: Parisi, Maria Florencia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; ArgentinaFil: Iriondo, Mirta Susana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Astronomia Teórica y Experimental. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba. Instituto de Astronomia Teórica y Experimental; ArgentinaFil: Cécere, Mariana Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; ArgentinaFil: Reula, Oscar Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentin