Problems In Computational Physics Research Articles

Optimised prefactored compact schemes for linear wave propagation phenomena

A family of space- and time-optimised prefactored compact schemes are developed that minimise the computational cost for given levels of numerical error in wave propagation phenomena, with special reference to aerodynamic sound. This work extends the approach of Pirozzoli [1] to the MacCormack type prefactored compact high-order schemes developed by Hixon [2], in which their shorter Padé stencil from the prefactorisation leads to a simpler enforcement of numerical boundary conditions. An explicit low-storage multi-step Runge–Kutta integration advances the states in time. Theoretical predictions for spatial and temporal error bounds are derived for the cost-optimised schemes and compared against benchmark schemes of current use in computational aeroacoustic applications in terms of computational cost for a given relative numerical error value. One- and two-dimensional test cases are presented to examine the effectiveness of the cost-optimised schemes for practical flow computations. An effectiveness up to about 50% higher than the standard schemes is verified for the linear one-dimensional advection solver, which is a popular baseline solver kernel for computational physics problems. A substantial error reduction for a given cost is also obtained in the more complex case of a two-dimensional acoustic pulse propagation, provided the optimised schemes are made to operate close to their nominal design points.

A Roadmap to Well Posed and Stable Problems in Computational Physics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

Quantum Simulator for Transport Phenomena in Fluid Flows.

Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.

Numerically exact three-dimensional propagation

Solving propagation in a three-dimensional environment is one of the most challenging problems in computational physics. However, for cases where the environment possesses some kind of symmetry be it rotational or translational, the propagation problem can be simplified considerably. It can be shown that in problems where the environment is translationally invariant, the 3D wave equation can be Fourier transformed along the direction of translational symmetry to reduce it to a 2D equation for each spectral component. The 3D solution can be obtained by solving the 2D wave equation for each spectral component and performing the inverse Fourier transform. In this paper, we use the above technique to compute propagation in an ideal and a penetrable wedge. For the ideal wedge, the pressure-release boundary condition is applied to both boundaries and for the penetrable wedge, the pressure-release boundary condition is applied to the horizontal surface and the continuity of pressure and normal velocity is imposed on the sloped interface. To obtain a numerically exact solution, we use the virtual source technique to solve the 2D problem for each spectral component.

A method to compute periodic sums

In a number of problems in computational physics, a finite sum of kernel functions centered at N particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension is usually treated via a different algorithm, Ewald summation, accelerated via the fast Fourier transform (FFT). A different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum into two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions (“sources”) placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via available summation algorithms. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating electrostatic potentials and forces. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at an additional cost of the order of the free-space FMM. Several technical details and efficient algorithms for auxiliary computations are provided, as are numerical comparisons.

A Survey on Parallel Computing and its Applications in Data-Parallel Problems Using GPU Architectures

AbstractParallel computing has become an important subject in the field of computer science and has proven to be critical when researching high performance solutions. The evolution of computer architectures (multi-coreandmany-core) towards a higher number of cores can only confirm that parallelism is the method of choice for speeding up an algorithm. In the last decade, the graphics processing unit, or GPU, has gained an important place in the field of high performance computing (HPC) because of its low cost and massive parallel processing power. Super-computing has become, for the first time, available to anyone at the price of a desktop computer. In this paper, we survey the concept of parallel computing and especially GPU computing. Achieving efficient parallel algorithms for the GPU is not a trivial task, there are several technical restrictions that must be satisfied in order to achieve the expected performance. Some of these limitations are consequences of the underlying architecture of the GPU and the theoretical models behind it. Our goal is to present a set of theoretical and technical concepts that are often required to understand the GPU and itsmassive parallelismmodel. In particular, we show how this new technology can help the field ofcomputational physics,especially when the problem isdata-parallel.We present four examples of computational physics problems;n-body, collision detection, Potts modelandcellular automatasimulations. These examples well represent the kind of problems that are suitable for GPU computing. By understanding the GPU architecture and its massive parallelism programming model, one can overcome many of the technical limitations found along the way, design better GPU-based algorithms for computational physics problems and achieve speedups that can reach up to two orders of magnitude when compared to sequential implementations.

Lattice differential operators for computational physics

We present a general scheme to derive lattice differential operators from the discrete velocities and associated Maxwell-Boltzmann distributions used in lattice hydrodynamics. Such discretizations offer built-in isotropy and lend themselves to recursive techniques to enhance the convergence order. The result is a simple and elegant procedure to derive isotropic and accurate discretizations of differential operators of general applicability across a broad range of problems in computational physics. We show the usefulness of this approach by providing examples from hydrodynamics and electrodynamics.

Linearly scaling direct method for accurately inverting sparse banded matrices

In many problems in computational physics and chemistry, one finds special kinds of sparse matrices, called banded matrices. These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, often need to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new algorithm for solving such a system, with the size of the matrix being n × n. We derive analytical recursive expressions that allow us to directly obtain the solution. In addition, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that our new algorithm is competitive, both in accuracy and in numerical efficiency, when compared with a standard method based on Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear in the calculation of Lagrange multipliers in proteins.

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Usually one can find three subjects in the first year of the syllabus of any technical engineering career, namely, calculus, general physics and programming. Being physics a matter lying on the grounds of technical engineering it becomes naturally appropriate to introduce the use of calculus and programming as useful tools in the context of a physics problem. This can be accomplished by moving some Practical Classes of Physics (problem solving) into the computer pool and by reformulating the physics problems in order to make them more appropriate for this kind of approach. In this environment, students put together, for instance, programming tools and numerical methods, along with the physical laws in order to address more realistic models, diferent from those which can usually be treated on the blackboard. This kind of computational physics problems increases the motivation of the engineering students by embedding them into sceneries whose models are closer to those real problems they will be facing later in their professional and scientific life. This is particularly relevant for the first year of the engineering careers when the development of this kind of professional skills is usually skipped. In the present work we will illustrate these ideas by means of the known problem of motion of a body subject to air drag force. The basic ideas of this work have been experienced in the physics course of first year undergraduate students of telecommunication and electronics engineering of Pinar del Rio University, Cuba in 2010.

A framework for parallel computational physics algorithms on multi-core: SPH in parallel

In this paper, a simulation framework that enables distributed numerical computing in multi-core shared-memory environments is presented. Using multiple threads allows a single memory image to be shared concurrently across cores but potentially introduces race conditions. Race conditions can be avoided by ensuring each core operates on an isolated memory block. This is usually achieved by running a different operating system process on each core, such as multiple MPI processes. However, we show that in many computational physics problems, memory isolation can also be enforced within a single process by leveraging spatial sub-division of the physical domain. A new spatial sub-division algorithm is presented that ensures threads operate on different memory blocks, allowing for in-place updates of state, with no message passing or creation of local variables during time stepping. Additionally, the developed framework controls task distribution dynamically ensuring an events based load balance. Results from fluid mechanics analysis using Smoothed Particle Hydrodynamics (SPH) are presented demonstrating linear performance with number of cores.

Canonical transforms and the efficient integration of quantum mechanical wave equations

The integration of time-dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The wave-function's energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations. We demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude. Our approach includes the so-called Kramers-Henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency.

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry, and biology. In this paper we use the first few eigenfunctions of the backward Fokker–Planck diffusion operator as a coarse-grained low dimensional representation for the long-term evolution of a stochastic system and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph...

Generalized perturbation-based stochastic finite element method in elastostatics

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.

On generalized stochastic perturbation‐based finite element method

AbstractGeneralized nth order stochastic perturbation technique, that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random parameters is proposed here. This technique is demonstrated in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of a bar. These numerical studies verify the influence of coefficient of variation of the random input and, at the same time, of the perturbation parameter on the first two probabilistic moments of the final solution vector. Copyright © 2005 John Wiley & Sons, Ltd.

Applications of Conformal Computing techniques to problems in computational physics: the Fast Fourier Transform

The techniques of Conformal Computing are introduced with an application to the Fast Fourier Transform. Conformal Computing is a design methodology, based on a rigorous mathematical foundation, which provides a systematic approach to the most efficient organization of all levels of the software and hardware design hierarchy from high-level software constructs all the way down to the design of the integrated circuits. We show that using these general design principles, without any specialized optimization, leads to portable, scalable, code that is competitive with other well-tuned machine specific routines. Further improvements are straightforward within our formalism by taking into account specific hardware details (e.g., cache loops) in a portable parametric way. We also argue that the present theory constitutes a uniform way of reasoning about physics and the data structures that define physics on computers.

A quasi‐analytical boundary condition for three‐dimensional finite difference electromagnetic modeling

Numerical modeling of the quasi‐static electromagnetic (EM) field in the frequency domain in a three‐dimensional (3‐D) inhomogeneous medium is a very challenging problem in computational physics. We present a new approach to the finite difference (FD) solution of this problem. The FD discretization of the EM field equation is based on the balance method. To compute the boundary values of the anomalous electric field we solve for, we suggest using the fast and accurate quasi‐analytical (QA) approximation, which is a special form of the extended Born approximation. We call this new condition a quasi‐analytical boundary condition (QA BC). This approach helps to reduce the size of the modeling domain without losing the accuracy of calculation. As a result, a larger number of grid cells can be used to describe the anomalous conductivity distribution within the modeling domain. The developed numerical technique allows application of a very fine discretization to the area with anomalous conductivity only because there is no need to move the boundaries too far from the inhomogeneous region, as required by the traditional Dirichlet or Neumann conditions for anomalous field with boundary values equal to zero. Therefore this approach increases the efficiency of FD modeling of the EM field in a medium with complex structure. We apply the QA BC and the traditional FD (with large grid and zero BC) methods to complicated models with high resistivity contrast. The numerical modeling demonstrates that the QA BC results in 5 times matrix size reduction and 2–3 times decrease in computational time.

Parallel Implementation of a Spectral Scheme for Simulations of 3-D Dynamic Fracture Events

The spectral scheme, whose parallel implementation is described in this paper, is currently one of the most efficient computational approaches available to solve a class of fundamental 3-D dynamic fracture problems. The scheme uses a special form of the boundary integral elastodynamic formulation and allows for the solution of the spontaneous motion (initiation, propagation, and arrest) of arbitrary shape planar cracks embedded in a homogeneous or bimaterial medium and subjected to arbitrary loading conditions. The parallel implementation described hereafter allows for the efficient solution of very large dynamic fracture problems, typically one to two orders of magnitudes larger than those available to date. It is based on a stacking scheme that distributes the spectral modes across the processors to achieve balance in memory and CPU time requirements. Although developed for dynamic fracture simulations, the spectral scheme and its parallel implementation are applicable to other problems in computational physics characterized by a convolution operation involving a decaying kernel.

Solving problems in computational physics using a general-purpose PDE solver

A general purpose partial differential equation solver, PDE2D, which is useful on a wide range of computational physics applications, including 1D, 2D and 3D, time-dependent, steady-state and eigenvalue problems, is described. This program has the advantages of being both accurate and very easy to use. Its wide applicability is illustrated by solving three representative quantum mechanics problems.

Barriers and challenges in computational fluid dynamics

Preface. 1: Accuracy. Progress in Applied Numerical Analysis for Computational Fluid Dynamics M.B. Giles. Examples of Error Propagation from Discontinuities B. Engquist, B. Sjogreen. Accurate Finite Difference Algorithms J.V. Goodrich. Computational Considerations for the Simulation of Discontinuous Flows M. H. Carpenter, J.H. Casper. Space-Time Methods for Hyperbolic Conservation Laws R.B. Lowrie, et al. II: Boundary Conditions and Stiffness Issues. Anisotropic Mesh Adaptation: A Step Towards a Mesh-Independent and User-Independent CFD W.G. Habashi, et al. Artificial Boundary Conditions for Infinite-Domain Problems S.V.Tsynkov. Issues and Strategies for Hyperbolic Problems with Stiff Source Terms M. Arora, P.L. Roe. III: Discontinuities. Numerical Methods for a One-Dimensional Interface Separating Compressible and Incompressible Flows R. Fedkiw, et al. On Some Outstanding Issues in CFD (1996) R. Lohner. Accurate and Robust Methods for Variable Density Incompressible Flows with Discontinuities W.J. Rider, et al. A Variational Approach to Deriving Smeared-Interface Surface Tension Models D. Jacqmin. IV: Other Applications. Compounded of Many Simples: Reflections on the Role of Model Problems in CFD P. Roe. A Unified CFD-Based Approach to a Variety of Problems in Computational Physics R.K. Agarwal. Second Order Godunov Schemes for 2D and 3D MHD Equations and Divergence-Free Condition Wenlong Dai, P.R. Woodward. On Multidimensional Positively Conservative High-Resolution Schemes T. Linde, P.L. Roe. A New Scheme for the Solutions of Multidimensional MHD Equations N. Aslan, T. Kamash . V: Convergence. Local Preconditioning: Who Needs It? B. Van Leer. The Quest for Diagonalization of Differential Systems P. Roe, E. Turkel. Multidimensional Upwinding: Unfolding the Mystery D. Sidilkover.

A SURVEY OF PARALLEL MULTIVARIATE NUMERICAL INTEGRATION: RECENT RESULTS AND DEVELOPMENTS

The main objective of this work is to present a survey of the most recently developed parallel multivariate integration routines for various architectures (MIMDDM, MIMD-SM, Network of workstations). Applications to computational physics problems is also provided.