Abstract The initial condition problem for a binary neutron star system requires a Poisson equation solver for the velocity potential with a Neumannlike boundary condition on the surface of the star. Difficulties that arise in this boundary value problem are: (a) the boundary is not known a priori , but constitutes part of the solution of the problem; (b) various terms become singular at the boundary. In this work, we present a new method to solve the fluid Poisson equation for irrotational/spinning binary neutron stars. The advantage of the new method is that it does not require complex fluid surface fitted coordinates and it can be implemented in a Cartesian grid, which is a standard choice in numerical relativity calculations. This is accomplished by employing the source term method proposed by Towers, where the boundary condition is treated as a jump condition and is incorporated as additional source terms in the Poisson equation, which is then solved iteratively. The issue of singular terms caused by vanishing density on the surface is resolved with an additional separation that shifts the computation boundary to the interior of the star. We present twodimensional tests to show the convergence of the source term method, and we further apply this solver to a realistic threedimensional binary neutron star problem. By comparing our solution with the one coming from the initial data solver cocal, we demonstrate agreement to approximately 1%. Our method can be used in other problems with nonsmooth solutions like in magnetized neutron stars.
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Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective
In the mean field integrateandfire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusionjump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the FokkerPlanck equation for the processwith random discharges (i.e. Poisson jumps) are defined on the whole space,while the equation for the limit process is on the half space. However, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomialorder convergence forthe distribution by a renormalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models.
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 NSFPAR ID:
 10355164
 Date Published:
 Journal Name:
 Mathematical Neuroscience and Applications
 Volume:
 Volume 1
 ISSN:
 28010159
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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