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Register a new userSelf-gravitational Force Calculation of High-Order Accuracy for Infinitesimally Thin Gaseous Disks
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Self-gravitational force calculation for infinitesimally thin disks is important for studies on the evolution of galactic and protoplanetary disks. Although high-order methods have been developed for hydrodynamic and magneto-hydrodynamic equations, high-order improvement is desirable for solving self-gravitational forces for thin disks. In this work, we present a new numerical algorithm that is of linear complexity and of high-order accuracy. This approach is fast since the force calculation is associated with a convolution form, and the fast calculation can be achieved using Fast Fourier Transform. The nice properties, such as the finite supports and smoothness, of B-splines are exploited to stably interpolate a surface density and achieve a high-order accuracy in forces. Moreover, if the mass distribution of interest is exclusively confined within a calculation domain, the method does not require artificial boundary values to be specified before the force calculation. To validate the proposed algorithm, a series of numerical tests, ranging from 1st- to 3rd-order implementations, are performed and the results are compared with analytic expressions derived for 3rd- and 4th-order generalized Maclaurin disks. We conclude that the improvement on the numerical accuracy is significant with the order of the method, with only little increase of the complexity of the method.
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Investigating the evolution of disk galaxies and the dynamics of proto-stellar disks can involve the use of both a hydrodynamical and a Poisson solver. These systems are usually approximated as infinitesimally thin disks using two- dimensional Cartesian or polar coordinates. In Cartesian coordinates, the calcu- lations of the hydrodynamics and self-gravitational forces are relatively straight- forward for attaining second order accuracy. However, in polar coordinates, a second order calculation of self-gravitational forces is required for matching the second order accuracy of hydrodynamical schemes. We present a direct algorithm for calculating self-gravitational forces with second order accuracy without artifi- cial boundary conditions. The Poisson integral in polar coordinates is expressed in a convolution form and the corresponding numerical complexity is nearly lin- ear using a fast Fourier transform. Examples with analytic solutions are used to verify that the truncated error of this algorithm is of second order. The kernel integral around the singularity is applied to modify the particle method. The use of a softening length is avoided and the accuracy of the particle method is significantly improved.
We extend the work of Yen et al. (2012) and develop 2nd order formulae to accommodate a nested grid discretization for the direct self-gravitational force calculation for infinitesimally thin gaseous disks. This approach uses a two-dimensional kernel derived for infinitesimally thin disks and is free of artificial boundary conditions. The self-gravitational force calculation is presented in generalized convolution forms for a nested grid configuration. A numerical technique derived from a fast Fourier transform is employed to reduce the computational complexity to be nearly linear. By comparing with analytic potential-density pairs associated with the generalized Maclaurin disks, the extended approach is verified to be of second order accuracy using numerical simulations. The proposed method is accurate, computationally fast and has the potential to be applied to the studies of planetary migration and the gaseous morphology of disk galaxies.
We present a simple and effective multigrid-based Poisson solver of second-order accuracy in both gravitational potential and forces in terms of the one, two and infinity norms. The method is especially suitable for numerical simulations using nested mesh refinement. The Poisson equation is solved from coarse to fine levels using a one-way interface scheme. We introduce anti-symmetrically linear interpolation for evaluating the boundary conditions across the multigrid hierarchy. The spurious forces commonly observed at the interfaces between refinement levels are effectively suppressed. We validate the method using two- and three-dimensional density-force pairs that are sufficiently smooth for probing the order of accuracy.
Self-force theory is the leading method of modeling extreme-mass-ratio inspirals (EMRIs), key sources for the gravitational-wave detector LISA. It is well known that for an accurate EMRI model, second-order self-force effects are critical, but calculations of these effects have been beset by obstacles. In this letter we present the first implementation of a complete scheme for second-order self-force computations, specialized to the case of quasicircular orbits about a Schwarzschild black hole. As a demonstration, we calculate the gravitational binding energy of these binaries.
In the supercritical range of the polytropic indices $gammain(1,frac43)$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows-up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case $gamma=1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.
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