Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity

Abstract

The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion.

Binary neutron-star mergers with Whisky and SACRA: First quantitative comparison of results from independent general-relativistic hydrodynamics codes

“…This operation produces two values at each cell boundary, which are then used as initial data for the local Riemann problems, whose (approximate) solution gives the fluxes through the cell boundaries. A method-of-lines approach [48], which reduces the partial differential equations (28) to a set of ordinary differential equations that can be evolved using standard numerical methods, such as Runge-Kutta or the iterative CranckNicholson schemes [49,50], is used to update the equations in time (see [15] for further details). Here, we employ the 4 th -order Runge-Kutta method (see below).…”

Relativistic g-modes in rapidly rotating neutron stars

“…Just like in that paper, a pseudospectral code is used to generate the axisymmetric background model which then gets interpolated onto a finite difference grid that extends the original computational domain in polar direction to 0 ≤ θ ≤ π. The time evolution equations themselves are solved by an Iterated Crank-Nicholson scheme with swapped weights ( [34,35]) and a weighting factor of α = 0.6. Similar to the barotropic case, the code is numerically unstable.…”

On iterated Crank–Nicolson methods for hyperbolic and parabolic equations

“…In particular, it was pointed out that stability properties of the original CN scheme are not approachable when the iteration number augments and that two iterations provide the largest stability and accuracy achievable in this iterative method [2]. In order to circumvent the stability shortcomings of the iterated CN method some generalizations were considered in [3,4] and respective analysis of stability for two-iteration schemes was performed in [5] both for advection and diffusion equations. Also another version of the iterated CN method was proposed in [5] and it was found that the last algorithm has better properties even providing only the first order of accuracy.…”

“…In order to circumvent the stability shortcomings of the iterated CN method some generalizations were considered in [3,4] and respective analysis of stability for two-iteration schemes was performed in [5] both for advection and diffusion equations. Also another version of the iterated CN method was proposed in [5] and it was found that the last algorithm has better properties even providing only the first order of accuracy. Different versions of the iterated CN method were used in numerical relativity by a number of authors (e.g., [3,4,[6][7][8] and references therein).…”