Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

The projection method is a widely used fractional-step algorithm for solving the incompressible Navier–Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier–Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.     

ELECTROSTATIC POTENTIAL OF STRONGLY NONLINEAR COMPOSITES: HOMOTOPY CONTINUATION APPROACH

The homotopy continuation method is used to solve the electrostatic boundary-value problems of strongly nonlinear composite media, which obey a current-field relation of J=σ E+χ|E|²E. With the mode expansion, the approximate analytical solutions of electric potential in host and inclusion regions are obtained by solving a set of nonlinear ordinary different equations, which are derived from the original equations with homotopy method. As an example in dimension two, we apply the method to deal with a nonlinear cylindrical inclusion embedded in a host. Comparing the approximate analytical solution of the potential obtained by homotopy method with that of numerical method, we can obverse that the homotopy method is valid for solving boundary-value problems of weakly and strongly nonlinear media.     

An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions

A highly efficient high-order boundary element method is developed for the numerical simulation of nonlinear wave–wave and wave-body interactions in the context of potential flow. The method is based on the framework of the quadratic boundary element method (QBEM) for the boundary integral equation and uses the pre-corrected fast Fourier transform (PFFT) algorithm to accelerate the evaluation of far-field influences of source and/or normal dipole distributions on boundary elements. The resulting PFFT–QBEM reduces the computational effort of solving the associated boundary-value problem from O(N^2～3) (with the traditional QBEM) to O(N ln N) where N represents the total number of boundary unknowns. Significantly, it allows for reliable computations of nonlinear hydrodynamics useful in ship design and marine applications, which are forbidden with the traditional methods on the presently available computing platforms. The formulation and numerical issues in the development and implementation of the PFFT–QBEM are described in detail. The characteristics of accuracy and efficiency of the PFFT–QBEM for various boundary-value problems are studied and compared to those of the existing accelerated (lower- and higher-order) boundary element methods. To illustrate the usefulness of the PFFT–QBEM, it is applied to solve the initial boundary-value problem in the generation of three-dimensional nonlinear waves by a moving ship hull. The predicted wave profile and resistance on the ship are compared to available experimental measurements with satisfactory agreements.     

A new formalism for time-dependent wave scattering from a bounded obstacle

A time-dependent three-dimensional acoustic scattering problem is considered. An incoming wave packet is scattered by a bounded, simply connected obstacle with locally Lipschitz boundary. The obstacle is assumed to have a constant boundary acoustic impedance. The limit cases of acoustically soft and acoustically hard obstacles are considered. The scattered acoustic field is the solution of an exterior problem for the wave equation. A new numerical method to compute the scattered acoustic field is proposed. This numerical method obtains the time-dependent scattered field as a superposition of time-harmonic acoustic waves and computes the time-harmonic acoustic waves by a new "operator expansion method." That is, the time-harmonic acoustic waves are solutions of an exterior boundary value problem for the Helmholtz equation. The method used to compute the time-harmonic waves improves on the method proposed by Misici, Pacelli, and Zirilli [J. Acoust. Soc. Am. 103, 106-113 (1998)] and is based on a "perturbative series" of the type of the one proposed in the operator expansion method by Milder [J. Acoust. Soc. Am. 89, 529-541 (1991)]. Computationally, the method is highly parallelizable with respect to time and space variables. Some numerical experiments on test problems obtained with a parallel implementation of the numerical method proposed are shown and discussed from the numerical and the physical point of view. The website: http://www.econ.unian.it/recchioni/w1 shows four animations relative to the numerical experiments.     

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.     

Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries

The pattern equations method is extended to solving the problems of wave scattering by bodies with piecewise smooth boundaries. The method is based on the reduction of the initial boundary-value problem to an integro-operator equation of the second kind in the scattering pattern of a body. With the use of the series expansion of the scattering pattern in angular spherical harmonics, the problem is ultimately reduced to solving an infinite algebraic system of equations in the expansion coefficients of the scattering pattern. The conditions at which this system can be solved by the method of reduction are formulated. Examples of solving the problems of wave scattering by bodies with impedance boundaries are considered. Essential advantages of the proposed method over other known methods are demonstrated.     

谐振管内非线性驻波的有限体积数值算法

为了扩展谐振管内非线性驻波在工程中的应用, 以及克服现有数值计算方法仅局限于求解直圆柱形和指数形谐振管内非线性驻波的问题. 根据变截面的非稳态可压缩热黏性流体Navier-Stokes方程和空间守恒方程, 并基于求解压力速度耦合方程的半隐式算法和交错网格技术, 构建一种能够计算任意形状轴对称谐振管受活塞驱动时内部非线性驻波的有限体积算法. 分别对圆柱形、指数形和圆锥形谐振管内的非线性驻波进行仿真计算. 通过与现有试验结果以及数值仿真结果的对比, 验证了该方法的正确性.并获得除驻波声压之外的另外一些新的物理结果, 包括速度、密度、温度的瞬时变化.在直圆柱形谐振管内产生冲击声压波, 速度波形中出现钉状结构.而在指数形和圆锥形谐振管内产生高声压幅值的驻波, 没有出现冲击波, 速度波形中均未发现钉状结构. 计算结果表明谐振管内非线性驻波的物理属性与谐振管形状之间有密切关系.     

Method of solving diffusion boundary-value problems for a region with a boundary moving in accordance with an arbitrary law

A general method is proposed for solving the boundary-value problem of the diffusion equation in a limited region with a boundary that moves in accordance with an arbitrary law. The method is used to solve the first linear diffusion problem. Other boundary-value problems can be solved in similar fashion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 97–101, December, 1970.     

解结构动力问题的半隐式Runge-Kutta型并行直接积分法

本文利用三级三阶半隐式Runge-Kutta法解结构动力问题,并用多项式预处理共轭梯度法解有关方程组。提出了半隐式Runge-Kutta型并行直接积分法RK33P。在YH-1机上,与相应的串行算法RK33S相比较,当有关方程组的阶数为10³~10⁴时,加速比可达24~27。     

Homotopy deform method for reproducing kernel space for nonlinear boundary value problems

In this paper, the combination of homotopy deform method (HDM) and simplified reproducing kernel method (SRKM) is introduced for solving the boundary value problems (BVPs) of nonlinear differential equations. The solution methodology is based on Adomian decomposition and reproducing kernel method (RKM). By the HDM, the nonlinear equations can be converted into a series of linear BVPs. After that, the simplified reproducing kernel method, which not only facilitates the reproducing kernel but also avoids the time-consuming Schmidt orthogonalization process, is proposed to solve linear equations. Some numerical test problems including ordinary differential equations and partial differential equations are analysed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that our algorithm is considerably accurate and effective as expected.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

We develop a parallel Jacobi–Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi–Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc’s efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrödinger’s equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi–Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.     

Improved MLPG_R method for simulating 2D interaction between violent waves and elastic structures

Interaction between violent water waves and structures is of a major concern and one of the important issues that has not been well understood in marine engineering. This paper will present first attempt to extend the Meshless Local Petrov Galerkin method with Rankine source solution (MLPG_R) for studying such interaction, which solves the Navier–Stokes equations for water waves and the elastic vibration equations for structures under wave impact. The MLPG_R method has been applied successfully to modeling various violent water waves and their interaction with rigid structures in our previous publications. To make the method robust for modeling wave elastic–structure interaction (hydroelasticity) problems concerned here, a near-strongly coupled and partitioned procedure is proposed to deal with coupling between violent waves and dynamics of structures. In addition, a novel approach is adopted to estimate pressure gradient when updating velocities and positions of fluid particles, leading to a relatively smoother pressure time history that is crucial for success in simulating problems about wave–structure interaction. The developed method is used to model several cases, covering a range from small wave to violent waves. Numerical results for them are compared with those obtained from other methods and from experiments in literature. Reasonable good agreement between them is achieved.     

A fast directional algorithm for high-frequency electromagnetic scattering

This paper is concerned with the fast solution of high-frequency electromagnetic scattering problems using the boundary integral formulation. We extend the O(N log N) directional multilevel algorithm previously proposed for the acoustic scattering case to the vector electromagnetic case. We also detail how to incorporate the curl operator of the magnetic field integral equation into the algorithm. When combined with a standard iterative method, this results in an almost linear complexity solver for the combined field integral equations. In addition, the butterfly algorithm is utilized to compute the far field pattern and radar cross section with O(N log N) complexity.     

Fast volumetric integral-equation solver for acoustic wave propagation through inhomogeneous media

Elements are described of a volumetric integral-equation-based algorithm applicable to accurate large-scale simulations of scattering and propagation of sound waves through inhomogeneous media. The considered algorithm makes possible simulations involving realistic geometries characterized by highly subwavelength details, large density contrasts, and described in terms of several million unknowns. The algorithm achieves its competitive performance, characterized by O(N log N) solution complexity and O(N) memory requirements, where N is the number of unknowns, through a fast and nonlossy fast Fourier transform based matrix compression technique, the adaptive integral method, previously developed for solving large-scale electromagnetic problems. Because of its ability of handling large problems with complex geometries, the developed solver may constitute an efficient and high fidelity numerical simulation tool for calculating acoustic field distributions in anatomically realistic models, e.g., in investigating acoustic energy transfer to the inner ear via nonairborne pathways in the human head. Examples of calculations of acoustic field distribution in a human head, which require solving linear systems of equations involving several million unknowns, are presented.     

New approach to solving the problem of composite-particle scattering on nuclei

An essentially new approach to solving the problem of elastic and inelastic scattering of a composite particle on stable nuclei is described. Within this approach, all channels of virtual breakup and stripping in the intermediate states are included in a nonlocal complex-valued interaction operator with the aid of the projection-operator technique.The three-particle continuum spectrum of the Hamiltonian for intermediate states in Q space is calculated within the orthogonalizing-pseudopotential method by introducing a pseudo-Hamiltonian, which is diagonalized in a full space in terms of a relevant oscillator basis. As was shown by a number of authors, the use of special quadratures makes it possible to reduce integration over the continuous spectrum of intermediate states to summation over a discretized continuum. On the basis of the formalism developed in this study, a closed Schrödinger equation with a nonlocal complex potential for partial waves is derived for describing elastic scattering of a composite particle by a target, and an explicit approximate formula for the amplitude of three-particle breakup is obtained on the same basis. This method has a number of obvious advantages over currently well-known approaches of the type of the discretized-continuum coupled-channel method, where solving the problem in question reduces to solving a cumbersome set of coupled equations.     

Accelerated Block Preconditioned Gradient method for large scale wave functions calculations in Density Functional Theory

An Accelerated Block Preconditioned Gradient (ABPG) method is proposed to solve electronic structure problems in Density Functional Theory. This iterative algorithm is designed to solve directly the non-linear Kohn–Sham equations for accurate discretization schemes involving a large number of degrees of freedom. It makes use of an acceleration scheme similar to what is known as RMM-DIIS in the electronic structure community. The method is illustrated with examples of convergence for large scale applications using a finite difference discretization and multigrid preconditioning.