11,991 research outputs found
Another approach to Runge-Kutta methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct optimization, avoiding the necessity to introduce simplifying assumptions upon the Runge-Kutta coefficients. More favourable coefficients, in view of rounding errors, are found
IMEX evolution of scalar fields on curved backgrounds
Inspiral of binary black holes occurs over a time-scale of many orbits, far
longer than the dynamical time-scale of the individual black holes. Explicit
evolutions of a binary system therefore require excessively many time steps to
capture interesting dynamics. We present a strategy to overcome the
Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern
implicit-explicit ODE solvers and multidomain spectral methods for elliptic
equations. Our analysis considers the model problem of a forced scalar field
propagating on a generic curved background. Nevertheless, we encounter and
address a number of issues pertinent to the binary black hole problem in full
general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild
coordinates, we document the results of several numerical experiments testing
our strategy.Comment: 28 pages, uses revtex4. Revised in response to referee's report. One
numerical experiment added which incorporates perturbed initial data and
adaptive time-steppin
Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach
A modified classical method for preliminary orbit determination is presented. In our proposal, the spread of the observations is considerably wider than in the original method, as well as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions, which will lead us to a class of iterative methods with a sixth local order of convergence. This is a process widely used in the design of iterative methods for solving nonlinear scalar equations, but rarely employed in vectorial cases. The numerical tests confirm the theoretical results, and the analysis of the dynamics of the problem shows the stability of the proposed schemes.The authors thank the anonymous referees for their valuable comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Andreu Estellés, C.; Cambil Teba, N.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach. Abstract and Applied Analysis. 2013. https://doi.org/10.1155/2013/960582S2013Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yBabajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., & Barati, A. (2008). A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule. Applied Mathematics and Computation, 200(1), 452-458. doi:10.1016/j.amc.2007.11.009Darvishi, M. T., & Barati, A. (2007). A third-order Newton-type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2), 630-635. doi:10.1016/j.amc.2006.08.080Darvishi, M. T., & Barati, A. (2007). Super cubic iterative methods to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(2), 1678-1685. doi:10.1016/j.amc.2006.11.022Cordero, A., Martínez, E., & Torregrosa, J. R. (2009). Iterative methods of order four and five for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 231(2), 541-551. doi:10.1016/j.cam.2009.04.015Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., Karami, A., & Barati, A. (2010). Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 233(8), 2002-2012. doi:10.1016/j.cam.2009.09.035Soleymani, F., Lotfi, T., & Bakhtiari, P. (2013). A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3), 1001-1015. doi:10.1007/s11590-013-0617-6Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Babajee, D. K. R., Cordero, A., Soleymani, F., & Torregrosa, J. R. (2012). On a Novel Fourth-Order Algorithm for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-12. doi:10.1155/2012/165452Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2012). Pseudocomposition: A technique to design predictor–corrector methods for systems of nonlinear equations. Applied Mathematics and Computation, 218(23), 11496-11504. doi:10.1016/j.amc.2012.04.081Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2013). Increasing the order of convergence of iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics, 252, 86-94. doi:10.1016/j.cam.2012.11.024Soleymani, F., & Stanimirović, P. S. (2013). A Higher Order Iterative Method for Computing the Drazin Inverse. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/708647Soleymani, F., Stanimirović, P. S., & Ullah, M. Z. (2013). An accelerated iterative method for computing weighted Moore–Penrose inverse. Applied Mathematics and Computation, 222, 365-371. doi:10.1016/j.amc.2013.07.039Sharma, J. R., Guha, R. K., & Sharma, R. (2012). An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numerical Algorithms, 62(2), 307-323. doi:10.1007/s11075-012-9585-7Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Abad, M. F., Cordero, A., & Torregrosa, J. R. (2013). Fourth- and Fifth-Order Methods for Solving Nonlinear Systems of Equations: An Application to the Global Positioning System. Abstract and Applied Analysis, 2013, 1-10. doi:10.1155/2013/586708Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zJarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015
High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates
High-order reconstruction schemes for the solution of hyperbolic conservation
laws in orthogonal curvilinear coordinates are revised in the finite volume
approach. The formulation employs a piecewise polynomial approximation to the
zone-average values to reconstruct left and right interface states from within
a computational zone to arbitrary order of accuracy by inverting a
Vandermonde-like linear system of equations with spatially varying
coefficients. The approach is general and can be used on uniform and
non-uniform meshes although explicit expressions are derived for polynomials
from second to fifth degree in cylindrical and spherical geometries with
uniform grid spacing. It is shown that, in regions of large curvature, the
resulting expressions differ considerably from their Cartesian counterparts and
that the lack of such corrections can severely degrade the accuracy of the
solution close to the coordinate origin. Limiting techniques and monotonicity
constraints are revised for conventional reconstruction schemes, namely, the
piecewise linear method (PLM), third-order weighted essentially non-oscillatory
(WENO) scheme and the piecewise parabolic method (PPM).
The performance of the improved reconstruction schemes is investigated in a
number of selected numerical benchmarks involving the solution of both scalar
and systems of nonlinear equations (such as the equations of gas dynamics and
magnetohydrodynamics) in cylindrical and spherical geometries in one and two
dimensions. Results confirm that the proposed approach yields considerably
smaller errors, higher convergence rates and it avoid spurious numerical
effects at a symmetry axis.Comment: 37 pages, 12 Figures. Accepted for publication in Journal of
Compuational Physic
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