The Convergence Estimation of the Parallel Algorithm of the Linear Cauchy Problem Solution for Large Systems of First-Order Ordinary Differential Equations Using the Solution as Expansion over Orthogonal Polynomials

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process.     

An improved algorithm and its parallel implementation for solving a general blood-tissue transport and metabolism model

Fast algorithms for simulating mathematical models of coupled blood-tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissues. Here, by combining the method of characteristics with the standard grid discretization technique, a novel algorithm is introduced for solving a general blood-tissue transport and metabolism model governed by a large system of one-dimensional semilinear first order partial differential equations. The key part of the algorithm is to approximate the model as a group of independent ordinary differential equation (ODE) systems such that each ODE system has the same size as the model and can be integrated independently. Thus the method can be easily implemented in parallel on a large-scale multiprocessor computer. The accuracy of the algorithm is demonstrated for solving a simple blood-tissue exchange model introduced by Sangren and Sheppard [W.C. Sangren, C.W. Sheppard, A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment, Bull. Math. Biophys. 15 (1953) 387–394], which has an analytical solution. Numerical experiments made on a distributed-memory parallel computer (an HP Linux cluster) and a shared-memory parallel computer (a SGI Origin 2000) demonstrate the parallel efficiency of the algorithm.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics, and a new algorithm—algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method. In the new algorithm, the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator. The exact analytical piece-like solution of the ordinary differential equations is expressed in terms of Taylor series with a local convergent radius, and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

Analytic-Numerical Construction of Nonstationary Probability Distributions for one Class of Nonlinear Stochastic Systems

We propose a method for constructing nonstationary model probability distributions for nonlinear dynamic systems related to the Verhulst stochastic equation. The proposed procedure is based on the numerical solution of relaxation differential equations for the mean and the variance. The set of the moment equations is closed and the probability density is constructed on the basis of rigorous analytical relations for the stationary probability characteristics. As a result, these distributions have correct stationary asymptotics. We show the possibility of numerical control of the accuracy of the proposed procedure. We consider the examples of relaxation of the probability characteristics of the amplitude of a self-oscillator and a parametric oscillator with a noise pump. The evolution of the amplitude probability distribution is found.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

Stability of Exponential Euler Method for Stochastic Systems under Poisson White Noise Excitations

The stability of stochastic systems under Poisson white noise excitations which based on the quantum theory is investigated in this paper. In general, the exact solution of the most of the stochastic systems with jumps is not easy to get. So it is very necessary to investigate the numerical solution of equations. On the one hand, exponential Euler method is applied to study stochastic delay differential equations, we can find the sufficient conditions for keeping mean square stability by investigating numerical method of systems. Through the comparison, we get the step-size of this method which is longer than the Euler-Maruyama method. On the other hand, mean square exponential stability of exponential Euler method for semi-linear stochastic delay differential equations under Poisson white noise excitations is confirmed.     

An alternative approach for the analysis of nonlinear vibrations of pipes conveying fluid

Based on a novel extended version of the Lagrange equations for systems containing non-material volumes, the nonlinear equations of motion for cantilever pipe systems conveying fluid are deduced. An alternative to existing methods utilizing Newtonian balance equations or Hamilton's principle is thus provided. The application of the extended Lagrange equations in combination with a Ritz method directly results in a set of nonlinear ordinary differential equations of motion, as opposed to the methods of derivation previously published, which result in partial differential equations. The pipe is modeled as a Euler elastica, where large deflections are considered without order-of-magnitude assumptions. For the equations of motion, a dimensional reduction with arbitrary order of approximation is introduced afterwards and compared with existing lower-order formulations from the literature. The effects of nonlinearities in the equations of motion are studied numerically. The numerical solutions of the extended Lagrange equations of the cantilever pipe system are compared with a second approach based on discrete masses and modeled in the framework of the multibody software HOTINT/MBS. Instability phenomena for an increasing number of discrete masses are presented and convergence towards the solution for pipes conveying fluid is shown.     

Concurrent implicit spectral deferred correction scheme for low-Mach number combustion with detailed chemistry

We present a parallel multi-implicit time integration scheme for the advection-diffusion-reaction systems arising from the equations governing low-Mach number combustion with complex chemistry. Our strategy employs parallelisation across the method to accelerate the serial Multi-Implicit Spectral Deferred Correction (MISDC) scheme used to couple the advection, diffusion, and reaction processes. In our approach, the diffusion solves and the reaction solves are performed concurrently by different processors. Our analysis shows that the proposed parallel scheme is stable for stiff problems and that the sweeps converge to the fixed-point solution at a faster rate than with serial MISDC. We present numerical examples to demonstrate that the new algorithm is high-order accurate in time, and achieves a parallel speedup compared to serial MISDC.     

Solution of the equation of radiative transfer using a Newton–Krylov approach and adaptive mesh refinement

The discrete ordinates method (DOM) and finite-volume method (FVM) are used extensively to solve the radiative transfer equation (RTE) in furnaces and combusting mixtures due to their balance between numerical efficiency and accuracy. These methods produce a system of coupled partial differential equations which are typically solved using space-marching techniques since they converge rapidly for constant coefficient spatial discretization schemes and non-scattering media. However, space-marching methods lose their effectiveness when applied to scattering media because the intensities in different directions become tightly coupled. When these methods are used in combination with high-resolution limited total-variation-diminishing (TVD) schemes, the additional non-linearities introduced by the flux limiting process can result in excessive iterations for most cases or even convergence failure for scattering media. Space-marching techniques may also not be quite as well-suited for the solution of problems involving complex three-dimensional geometries and/or for use in highly-scalable parallel algorithms. A novel pseudo-time marching algorithm is therefore proposed herein to solve the DOM or FVM equations on multi-block body-fitted meshes using a highly scalable parallel-implicit solution approach in conjunction with high-resolution TVD spatial discretization. Adaptive mesh refinement (AMR) is also employed to properly capture disparate solution scales with a reduced number of grid points. The scheme is assessed in terms of discontinuity-capturing capabilities, spatial and angular solution accuracy, scalability, and serial performance through comparisons to other commonly employed solution techniques. The proposed algorithm is shown to possess excellent parallel scaling characteristics and can be readily applied to problems involving complex geometries. In particular, greater than 85% parallel efficiency is demonstrated for a strong scaling problem on up to 256 processors. Furthermore, a speedup of a factor of at least two was observed over a standard space-marching algorithm using a limited scheme for optically thick scattering media. Although the time-marching approach is approximately four times slower for absorbing media, it vastly outperforms standard solvers when parallel speedup is taken into account. The latter is particularly true for geometrically complex computational domains.     

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.     

印制线抗电磁干扰模型分析与计算

对双导体传输线的Taylor模型在频域下的非齐次线性微分方程组,采用常数变异法求得其在边界条件下解的表达式,所求得传输线在终端响应的电压和电流值与BLT方程求得的结果相同。同时,还得到了印刷电路板上的平行双线在外电磁场辐射下终端响应电压的解析表达式,并通过解析解分析得到了不产生干扰电压时的电磁波的入射角度和极化方向。     

Numerical methods for solving one-dimensional cochlear models in the time domain

In this article, a robust numerical solution method for one-dimensional (1-D) cochlear models in the time domain is presented. The method has been designed particularly for models with a cochlear partition having nonlinear and active mechanical properties. The model equations are discretized with respect to the spatial variable by means of the principle of Galerkin to yield a system of ordinary differential equations in the time variable. To solve this system, several numerical integration methods concerning stability and computational performance are compared. The selected algorithm is based on a variable step size fourth-order Runge-Kutta scheme; it is shown to be both more stable and much more efficient than previously published numerical solution techniques.     

Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

High resolution and scalable parallel algorithms for the shallow water equations on the sphere are very important for modeling the global climate. In this paper, we introduce and study some highly scalable multilevel domain decomposition methods for the fully implicit solution of the nonlinear shallow water equations discretized with a second-order well-balanced finite volume method on the cubed-sphere. With the fully implicit approach, the time step size is no longer limited by the stability condition, and with the multilevel preconditioners, good scalabilities are obtained on computers with a large number of processors. The investigation focuses on the use of semismooth inexact Newton method for the case with nonsmooth topography and the use of two- and three-level overlapping Schwarz methods with different order of discretizations for the preconditioning of the Jacobian systems. We test the proposed algorithm for several benchmark cases and show numerically that this approach converges well with smooth and nonsmooth bottom topography, and scales perfectly in terms of the strong scalability and reasonably well in terms of the weak scalability on machines with thousands and tens of thousands of processors.     

不可压缩流的并行两水平稳定有限元算法

提出一种数值求解定常不可压缩Stokes方程的并行两水平Grad-div稳定有限元算法。首先在粗网格中求解Grad-div稳定化的全局解, 再在相互重叠的细网格子区域上并行纠正。通过对稳定化参数、粗细网格尺寸恰当的选取, 该方法可得到最优收敛率, 数值结果验证了算法的高效性。     

Influence of the opening angle of a conical supersonic nozzle on the structure of initial interval of non-isobaric jet

The ideal gas exhaustion from an infinite volume into a gas at rest through a supersonic conical Laval nozzle is considered. The problem was solved numerically by steadying in time in a unified formulation for the regions inside the nozzle and in the ambient environment. In such a statement, the nozzle outlet section is no internal boundary of the region under consideration, and there is no need of specifying the boundary conditions here. Local subsonic zones arising in the flow lie inside the region under consideration, which eliminates the possibility of using a marching technique along one of the coordinates. The numerical solution is constructed by a unified algorithm for the entire flow region, which gives a possibility of obtaining a higher accuracy. The computations are carried out in the jet initial interval, where, according to monograph [1], the wave phenomena predominate over the viscous effects. The exhaustion process is described by the system of gas dynamics equations. Their solution is constructed with the aid of a finite difference Harten’s TVD (Total Variation Diminishing) scheme [2], which has the second approximation order in space. The second approximation order in time is achieved with the aid of a five-stage Runge-Kutta method. The solution algorithm has been parallelized in space and implemented on the multi-processor computer systems of the ITAM SB RAS and the MVS-128 of the Siberian Supercomputer Center of SB RAS. The influence of the semi-apex angle of the nozzle supersonic part and the pressure jump between the nozzle outlet section and the ambient environment on the flow in the initial interval of a non-isobaric jet is investigated in the work. A comparison with experimental data is presented. The computations are carried out for the semi-apex angles of the nozzle supersonic part from 0 (parallel flow) to 20 degrees. For all considered nozzles, the Mach number in the nozzle outlet section, which was computed from the one-dimensional theory, equaled three. Computations showed that in the case of flow acceleration in a conical supersonic nozzle, its geometry is one of the main factors determining the formation of the jet initial interval in ambient environment.     

Gröbner bases applied to systems of linear difference equations

In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high-energy physics as recurrence relations for multiloop Feynman integrals. The most universal algorithmic tool for investigation of linear difference systems is based on their transformation into an equivalent basis form. We present an algorithm for this transformation implemented in Maple. The algorithm and its implementation can be applied to automatic generation of difference schemes for linear partial differential equations and to reduction of Feynman integrals. Some illustrative examples are given. The text was submitted by the author in English.     

Dimension reduction method for ODE fluid models

We develop a new dimension reduction method for large size systems of ordinary differential equations (ODEs) obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large-size classical particle systems with negligibly small variations of particle concentration. We propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution. To illustrate the computational advantages of the proposed reduction method, we use it to solve a system of smoothed particle hydrodynamic ODEs describing single-phase and two-phase layered Poiseuille flows driven by uniform and periodic (in space) body forces. For the single-phase Poiseuille flow driven by the uniform force, the coarse solution was obtained with the zero-order deconvolution. For the single-phase flow driven by the periodic body force and for the two-phase flows, the higher-order (the first- and second-order) deconvolutions were necessary to obtain a sufficiently accurate solution.     

High-order quadratures for the solution of scattering problems in two dimensions

We construct an iterative algorithm for the solution of forward scattering problems in two dimensions. The scheme is based on the combination of high-order quadrature formulae, fast application of integral operators in Lippmann–Schwinger equations, and the stabilized bi-conjugate gradient method (BI-CGSTAB). While the FFT-based fast application of integral operators and the BI-CGSTAB for the solution of linear systems are fairly standard, a large part of this paper is devoted to constructing a class of high-order quadrature formulae applicable to a wide range of singular functions in two and three dimensions; these are used to obtain rapidly convergent discretizations of Lippmann–Schwinger equations. The performance of the algorithm is illustrated with several numerical examples.