Abstract The initial condition problem for a binary neutron star system requires a Poisson equation solver for the velocity potential with a Neumann-like 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 two-dimensional tests to show the convergence of the source term method, and we further apply this solver to a realistic three-dimensional 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 non-smooth solutions like in magnetized neutron stars.
more »
« less
Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective
In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump 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 Fokker-Planck 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 polynomial-order convergence forthe distribution by a re-normalization 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.
more »
« less
- NSF-PAR ID:
- 10355164
- Date Published:
- Journal Name:
- Mathematical Neuroscience and Applications
- Volume:
- Volume 1
- ISSN:
- 2801-0159
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.more » « less
-
This paper develops a tree-topological local mesh refinement (TLMR) method on Cartesian grids for the simulation of bio-inspired flow with multiple moving objects. The TLMR nests refinement mesh blocks of structured grids to the target regions and arrange the blocks in a tree topology. The method solves the time-dependent incompressible flow using a fractional-step method and discretizes the Navier-Stokes equation using a finite-difference formulation with an immersed boundary method to resolve the complex boundaries. When iteratively solving the discretized equations across the coarse and fine TLMR blocks, for better accuracy and faster convergence, the momentum equation is solved on all blocks simultaneously, while the Poisson equation is solved recursively from the coarsest block to the finest ones. When the refined blocks of the same block are connected, the parallel Schwarz method is used to iteratively solve both the momentum and Poisson equations. Convergence studies show that the algorithm is second-order accurate in space for both velocity and pressure, and the developed mesh refinement technique is benchmarked and demonstrated by several canonical flow problems. The TLMR enables a fast solution to an incompressible flow problem with complex boundaries or multiple moving objects. Various bio-inspired flows of multiple moving objects show that the solver can save over 80% computational time, proportional to the grid reduction when refinement is applied.more » « less
-
We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.more » « less
-
Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46298/mna.7203
