ﻻ يوجد ملخص باللغة العربية
In most of mesh-free methods, the calculation of interactions between sample points or particles is the most time consuming. When we use mesh-free methods with high spatial orders, the order of the time integration should also be high. If we use usual Runge-Kutta schemes, we need to perform the interaction calculation multiple times per one time step. One way to reduce the number of interaction calculations is to use Hermite schemes, which use the time derivatives of the right hand side of differential equations, since Hermite schemes require smaller number of interaction calculations than RK schemes do to achieve the same order. In this paper, we construct a Hermite scheme for a mesh-free method with high spatial orders. We performed several numerical tests with fourth-order Hermite schemes and Runge-Kutta schemes. We found that, for both of Hermite and Runge-Kutta schemes, the overall error is determined by the error of spatial derivatives, for timesteps smaller than the stability limit. The calculation cost at the timestep size of the stability limit is smaller for Hermite schemes. Therefore, we conclude that Hermite schemes are more efficient than Runge-Kutta schemes and thus useful for high-order mesh-free methods for Lagrangian Hydrodynamics.
Recently, a 4th-order asymptotic preserving multiderivative implicit-explicit (IMEX) scheme was developed (Schutz and Seal 2020, arXiv:2001.08268). This scheme is based on a 4th-order Hermite interpolation in time, and uses an approach based on opera
We present sixth- and eighth-order Hermite integrators for astrophysical $N$-body simulations, which use the derivatives of accelerations up to second order ({it snap}) and third order ({it crackle}). These schemes do not require previous values for
We present a family of modified Hermite integrators of arbitrary order possessing superior behaviour for the integration of Keplerian and near-Keplerian orbits. After recounting the derivation of Hermite N-body integrators of arbitrary order, we deri
We develop new numerical schemes for Vlasov--Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that positivity of the distribution function is also preser
Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack o