Stable second-order scheme for integrating the Kuramoto-Sivanshinsky equation in polar coordinates using distributed approximating functionals

2005 ◽  
Vol 72 (3) ◽  
Author(s):  
Peter Blomgren ◽  
Scott Gasner ◽  
Antonio Palacios
Keyword(s):  
Second Order ◽  
Polar Coordinates ◽  
Distributed Approximating Functionals ◽  
Order Scheme

scholarly journals Beyond second-order convergence in simulations of magnetized binary neutron stars with realistic microphysics

2019 ◽  
Vol 490 (3) ◽  
pp. 3588-3600 ◽  
Author(s):  
E R Most ◽  
L Jens Papenfort ◽  
L Rezzolla
Keyword(s):  
Neutron Stars ◽  
Fourth Order ◽  
Finite Temperature ◽  
Equations Of State ◽  
Magnetic Energy ◽  
Second Order ◽  
Conservative Finite Difference Scheme ◽  
Order Scheme ◽  
Convergence Orders ◽  
The Impact

ABSTRACT We investigate the impact of using high-order numerical methods to study the merger of magnetized neutron stars with finite-temperature microphysics and neutrino cooling in full general relativity. By implementing a fourth-order accurate conservative finite-difference scheme we model the inspiral together with the early post-merger and highlight the differences to traditional second-order approaches at the various stages of the simulation. We find that even for finite-temperature equations of state, convergence orders higher than second order can be achieved in the inspiral and post-merger for the gravitational-wave phase. We further demonstrate that the second-order scheme overestimates the amount of proton-rich shock-heated ejecta, which can have an impact on the modelling of the dynamical part of the kilonova emission. Finally, we show that already at low resolution the growth rate of the magnetic energy is consistently resolved by using a fourth-order scheme.

scholarly journals On certain Quadric Hypersurfaces in Riemannian Space

1935 ◽  
Vol 4 (2) ◽  
pp. 85-91 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Riemannian Space ◽  
Second Order ◽  
Polar Coordinates ◽  
Intrinsic Geometry ◽  
Constant Distance

The use of geodesic polar coordinates in the intrinsic geometry of a surface leads to the concept of a geodesic circle, i.e. the locus of points at a constant distance from the pole 0 along the geodesics through 0. A geodesic hypersphere is the obvious generalisation of this for a Riemannian Vn. We propose to consider more general central quadric hypersurfaces of Vn, which we define as follows. Let xi (i = 1, 2, …, n) be a system of coordinates in Vn, whose metric is gijdxidxj, and let aij be the components in the x's of a symmetric covariant tensor of the second order, evaluated at the point 0, which is taken as pole.

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