Search

Abstract Large computation time is widely considered to be the most important issue in scientific research especially in solving structural evolution problems. Recent developments in this domain have shown that the use of non-incremental schemes through Model Order Reduction led to important results in saving time. Yet, the question arises here how to obtain more time-saving. This paper examines an approach based on a collection of significant modes given by Proper Generalized Decomposition (PGD) solution for different time scales in order to save more computation time. The dictionary of the significant modes allows to construct an accurate solution for different characteristic times and different boundary problems compared to the full solution with a relative error rate less than 5% and with a large time saving of order 50 compared to Finite Element Method (FEM). The ability of the approach with respect to cycle time is discussed. Highlights • A new approach is proposed to decrease computational time for cyclic problems involving different time scales. • The approach is based on the use of pre-computed space–time modes which are further collected in a dictionary. • For a physical time belonging to the range of the dictionary, the solution can be accurately predicted. • For a given cycle time, the dictionary leads to an accurate if it contains the modes associated with this specific time. [ABSTRACT FROM AUTHOR]

The singularity in the traditional spherical polar coordinate system at the poles is a major factor in the lack of scalability of atmospheric models on massively parallel machines. Overset grids such as the Yin-Yang grid introduced by Kageyama and Sato [1] offer a potential solution to this problem. In this paper a three-dimensional, compressible, non-hydrostatic atmospheric model is developed and tested on the Yin-Yang grid building on ideas previously developed by the authors on the solution of Elliptic boundary value problems and conservation on overset grids. Using several tests from the literature, it is shown that this model is highly stable (even with little off-centering), accurate, and highly efficient in terms of computational cost. The model also incorporates highly efficient and accurate approaches to achieve positivity, monotonicity and conservative transport, which are paramount requirements for any atmospheric model. The parallel scalability of this model, using in excess of 212 million unknowns and more than 6000 processors, is also discussed and shown to compare favourably with a highly optimised latitude–longitude model in terms of scalability and actual run times. [ABSTRACT FROM AUTHOR]

Abstract: In this first part of a series of two papers, we present in considerable detail a collision-driven molecular dynamics algorithm for a system of non-spherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. The algorithm extends previous event-driven molecular dynamics algorithms for spheres, and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size. We present a novel partial-update near-neighbor list (NNL) algorithm that is superior to previous algorithms at high densities, without compromising the correctness of the algorithm. This efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes (BSC). These techniques will be useful in any particle-based simulation, including Monte Carlo and time-driven molecular dynamics. Additionally, we allow for a non-vanishing rate of deformation of the boundary, which can be used to model macroscopic strain and also alleviate boundary effects for small systems. In the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation, demonstrating the practical utility of the algorithm. [Copyright &y& Elsevier]

In computations, it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer (PML) of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation. Unfortunately, the PML does not give us an indication about appropriate boundary conditions needed to close the edges of the PML, or how those boundary conditions should be enforced in a numerical setting. Terminating the PML with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the PML which will eventually destroy the accuracy of a numerical simulation everywhere. In this paper, we analyze the stability and the well-posedness of boundary conditions terminating the PML for the elastic wave equation in first order form. First, we consider a vertical modal PML truncating a two space dimensional computational domain in the horizontal direction. We freeze all coefficients and consider a left half-plane problem with linear boundary conditions terminating the PML. The normal mode analysis is used to study the stability and well-posedness of the resulting initial boundary value problem (IBVP). The result is that any linear well-posed boundary condition yielding an energy estimate for the elastic wave equation, without the PML, will also lead to a well-posed IBVP for the PML. Second, we extend the analysis to the PML corner region where both a horizontal and vertical PML are simultaneously active. The challenge lies in constructing accurate and stable numerical approximations for the PML and the boundary conditions. Third, we develop a high order accurate finite difference approximation of the PML subject to the boundary conditions. To enable accurate and stable numerical boundary treatments for the PML we construct continuous energy estimates in the Laplace space for a one space dimensional problem and two space dimensional PML corner problem. We use summation-by-parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. In order to ensure numerical stability of the discrete PML, it is necessary to extend the numerical boundary procedure to the auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical experiments are presented corroborating the theoretical results. Moreover, in order to ensure longtime numerical stability, the boundary condition closing the PML, or its corresponding discrete implementation, must be dissipative. Furthermore, the numerical experiments demonstrate the stable and robust treatment of PML corners. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge–Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit. [Copyright &y& Elsevier]

Abstract: The method of fundamental solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper, we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an implicit high-order compact (HOC) finite-difference scheme for solving the two-dimensional (2D) unsteady Navier–Stokes (N–S) equations on irregular geometries on orthogonal grids. Our scheme is second order accurate in time and fourth order accurate in space. It is used to solve three pertinent fluid flow problems, namely, the flow decayed by viscosity, the lid-driven square cavity and the flow in a constricted channel. It is seen to efficiently capture both transient and steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. Apart from including the good features of HOC schemes, our formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Detailed comparison data produced by the scheme for all the three test cases are provided and compared with analytical as well as established numerical results. Excellent comparison is obtained in all the cases. [Copyright &y& Elsevier]