We present a formula approximating the mean escape time (MST) of a particle from a tilted multi-periodic potential well. The potential function consists of a weighted sum of a finite number of component functions, each of which is periodic. For this particular case, the least period of the potential function is a common period amongst all of its component functions. An approximation of the MST for the potential function is derived, and this approximation takes the form of a product of the MSTs for each of the individual periodic component functions. Our first example illustrates the computational advantages of using the approximation for model validation and parameter tuning in the context of the biological application of DNA transcription. We also use this formula to approximate the MST for an arbitrary tilted periodic potential by the product of MSTs of a finite number of its Fourier modes. Two examples using truncated Fourier series are presented and analyzed.

A local radial basis function method (LRBF) is applied for the solution of boundary value problems in annular domains governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity. By appropriately choosing the collocation points we obtain linear systems in which the coefficient matrices possess block sparse circulant structures and which can be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). The MDAs used are appropriately modified to take into account the sparsity of the arrays involved in the discretization. The leave-one-out cross validation (LOOCV) algorithm is employed to obtain a suitable value for the shape parameter in the radial basis functions (RBFs) used. The selection of the nearest centres for each local influence domain is carried out using a modification of the kd-tree algorithm. In several numerical experiments, it is demonstrated that the proposed algorithm is both accurate and capable of solving large scale problems.

In this study, an N-body simulation code was developed for self-gravitating systems with a limited first-order post-Newtonian approximation. The code was applied to a special case in which the system consists of one massive object and many low-mass objects. Therefore, the behavior of stars around the massive black hole could be analyzed. A graphics processing unit (GPU) was used to accelerate the code execution, and it could be accelerated by several tens of times compared to a single-core CPU for N ≃10⁴ objects.

In this paper, a conformal energy-conserved scheme is proposed for solving the Maxwell's equations with the perfectly matched layer. The equations are split as a Hamiltonian system and a dissipative system, respectively. The Hamiltonian system is solved by an energy-conserved method and the dissipative system is integrated exactly. With the aid of the Strang splitting, a fully-discretized scheme is obtained. The resulting scheme can preserve the five discrete conformal energy conservation laws and the discrete conformal symplectic conservation law. Based on the energy method, an optimal error estimate of the scheme is established in discrete L²-norm. Some numerical experiments are addressed to verify our theoretical analysis.

In this paper we present an extended formulation of the immersed boundary (IB) method that facilitates simulation of incompressible immiscible two-phase flows. In the developed formulation the pressure field and the surface tension forces associated with interface curvature are implicitly introduced in the form of distributed Lagrange multipliers. The approach provides for impermeability between both phases and exhibits accurate mass conservation without the need for additional correction procedures. Further, we present a grid independence study and extensive verification of the developed method for representative 2D two-phase flows dominated by buoyancy, shear stress, and surface tension forces.

In this paper, we first develop the mathematical modeling equations for wave propagation in several transformation optics devices, including electromagnetic concentrator, rotator and splitter. Then we propose the corresponding finite element time-domain methods for simulating wave propagation in these transformation optics devices. We implement the proposed algorithms and our numerical results demonstrate the effectiveness of our modeling equations. To our best knowledge, this is the first work on time-domain finite element simulation carried out for the electromagnetic concentrator, rotator and splitter.

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets. The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and an anisotropic weak anchoring energy that enforces a condition such that the director field is aligned perpendicular to the interface of the droplet. Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics. We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy, as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part. Next, we present a discrete gradient flow method and show that it is monotone energy decreasing. Furthermore, we show that global discrete energy minimizers Γ-converge to global minimizers of the continuous energy. We conclude with numerical experiments illustrating different gradient flow dynamics, including droplet coalescence and break-up.

This paper presents an extension work of the hybrid scheme proposed by Wang et al. [J. Comput. Phys. 229 (2010) 169-180] for numerical simulation of subsonic isotropic turbulence to supersonic turbulence regime. The scheme still utilizes an 8th-order compact scheme with built-in hyperviscosity for smooth regions and a 7th-order WENO scheme for highly compression regions, but now both in their conservation formulations and for the latter with the Roe type characteristic-wise reconstruction. To enhance the robustness of the WENO scheme without compromising its high-resolution and accuracy, the recursive-order-reduction procedure is adopted, where a new type of reconstruction-failure-detection criterion is constructed from the idea of positivity-preserving. In addition, a new form of cooling function is proposed, which is proved also to be positivity-preserving. With a combination of these techniques, the new scheme not only inherits the good properties of the original one but also extends largely the computable range of turbulent Mach number, which has been further confirmed by numerical results.

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h²+τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

In this paper, we introduce a physics-based nonlinear preconditioned Inexact Newton Method (INB) for the multi-physical simulation of fractured reservoirs. Instead of solving the partial differential equations (PDE) exactly, Inexact Newton method finds a direction for the iteration and solves the equations inexactly with fewer iterations. However, when the equations are not smooth enough, especially when local discontinuities exits, and when proper preconditioning operations are not adopted, the Inexact Newton method may be slow or even stagnant.
As pointed out by Keyes et al. [1], multi-physical numerical simulation faces several challenges, one of which is the local-scale nonlinearity and discontinuity. In this work, we have proposed and studied a nonlinear preconditioner to improve the performance of Inexact Newton Method. The nonlinear preconditioner is essentially a physics-based strategy to adaptively identify and eliminate the highly nonlinear zones.
The proposed algorithm has been implemented into our fully coupled, fully implicit THM reservoir simulator (Wang et al. [2, 3]) to study the effects of cold water injection on fractured petroleum reservoirs. The results of this work show that after the implementation of this nonlinear preconditioner, the iterative solver has become significantly more robust and efficient.

The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the efficiency of the proposed method.

We investigate the statistical behaviors of two-dimensional dry foam using numerical simulations based on the immersed boundary (IB) method. We model the liquid phase of a foam as a thin elastic boundary with surface tension and the gas phase as a viscous incompressible fluid which can go through the liquid boundary. We present evidence of the existence of a limiting scaling state of the dry foam dynamics in which the asymptotic value of µ₂, the second moment of the distribution of the numbers of cell sides, lies in the range of 1.3±0.3. We also numerically verify some well-known formulas derived in the dynamics of two-dimensional dry foam such as von Neumann relation, Lewis law, and Aboav-Weaire law. Our simulation results are comparable to those of soap froth experiments and Potts model simulations. Furthermore, we investigate the statistical behaviors of two-dimensional dry foam in an oscillatory shear flow to show the applicability of our method to more general flow conditions.