Algorithms For Ordinary Differential Equations Research Articles

常微分方程常用的数值算法及其应用

常微分方程常用的数值算法及其应用

A New Parareal Algorithm for Problems with Discontinuous Sources

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary differential equations which involve discontinuous right-hand sides. Such situations occur in various applications, e.g., when an electric device is supplied with a pulse-width-modulated signal. Our new Parareal algorithm uses a smooth input for the coarse problem with reduced dynamics. We derive error estimates that show how the input reduction influences the overall convergence rate of the algorithm. We support our theoretical results by numerical experiments, and also test our new Parareal algorithm in an eddy current simulation of an induction machine.

Water Hammer Peak Pressures and Decay Rates of Transients in Smooth Lines With Turbulent Flow

Transient pressure peak values and decay rates associated with water hammer surges in fluid lines are investigated using an analytical method that has been formulated, in a previous publication, to simulate pressure transients in turbulent flow. The method agrees quite well with method of characteristics (MOC) simulations of unsteady friction models and has been verified with experimental data available for Reynolds numbers out to 15,800. The method is based on the formulation of ordinary differential equations from the frequency response of a pressure transfer function using an inverse frequency algorithm. The model is formulated by dividing the line into n-sections to distribute the turbulence resistance along the line at higher Reynolds numbers. In this paper, it will be demonstrated that convergence of the analytical solution is achieved with as few as 5–10 line sections for Reynolds numbers up to 200,000. The method not only provides for the use of conventional time domain solution algorithms for ordinary differential equations but also provides empirical equations for estimating peak surge pressures and transient decay rates as defined by eigenvalues. For typical sets of line and fluid properties, the trend of the damping ratio of the first or dominate mode of the pressure transients transfer function is found to be an approximate linear function of a dimensionless parameter that is a function of the Reynolds number. In addition, a reasonably accurate dimensionless trend formula for estimates of the normalized peak pressures is formulated and presented.

Symmetry Reduction of Ordinary Differential Equations Using Moving Frames

The symmetry reduction algorithm for ordinary differential equations due to Sophus Lie is revisited using the method of equivariant moving frames. Using the recurrence formulas provided by the theory of equivariant moving frames, computations are performed symbolically without relying on the coordinate expressions for the canonical variables and the differential invariants occurring in Lie’s original procedure.

Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model

We present a generalisation of the mathematical models describing the interactions between the immune system and tumour cells which takes into account distributed time delays. For the analytical study we do not assume any particular form of the stimulus function describing the immune system reaction to presence of tumour cells but we only postulate its general properties. We analyse basic mathematical properties of the considered model such as existence and uniqueness of the solutions. Next, we discuss the existence of the stationary solutions and analytically investigate their stability depending on the forms of considered probability densities that is: Erlang, triangular and uniform probability densities separated or not from zero. Particular instability results are obtained for a general type of probability densities. Our results are compared with those for the model with discrete delays know from the literature. In addition, for each considered type of probability density, the model is fitted to the experimental data for the mice B-cell lymphoma showing mean square errors at the same comparable level. For estimated sets of parameters we discuss possibility of stabilisation of the tumour dormant steady state. Instability of this steady state results in uncontrolled tumour growth. In order to perform numerical simulation, following the idea of linear chain trick, we derive numerical procedures that allow us to solve systems with considered probability densities using standard algorithm for ordinary differential equations or differential equations with discrete delays.

Alternative Approach for Modeling Transients in Smooth Pipe With Low Turbulent Flow

A new simplified approach for modeling and simulating pressure transients resulting from the rapid acceleration or deceleration of turbulent flow in smooth-walled fluid lines is introduced. In contrast to previous approaches for modeling turbulence by modifying the head loss terms in the momentum partial differential equation, this approach is achieved by coupling the frequency domain analytical solution to the laminar flow version of the partial differential equations in series with a lumped resistance that has been sized so that the steady flow resistance for the line is equivalent to an empirical turbulent steady flow resistance. The model provides normalized pressure and flow transients that have good agreement with experimental data and with method-of-characteristics (MOC) solutions associated with previously validated turbulence models. The motivation for this research is based on the need for a practical means to simulate the effects of fluid transients in lines that are internal components within a total engineering system without the need to understand the different unsteady turbulence one- and two-dimensional (1D/2D) friction models and also be proficient with the complexities of nonlinear interaction of friction and interpolation errors encountered using MOC. This modeling approach utilizes a preprogramed inverse frequency algorithm, commonly used for system identification, which generates “equivalent” high-order normalized linear ordinary differential equations that can be coupled with models for other fluid power components and easily solved in the time domain using preprogramed numerical algorithms for ordinary differential equations.

Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

We propose new Legendre–Gauss collocation algorithms for ordinary differential equations. We also design Legendre–Gauss-type collocation algorithms for time-dependent nonlinear partial differential equations. The suggested algorithms enjoy spectral accuracy in both time and space, and can be implemented in a fast and stable manner. Numerical results exhibit the effectiveness.

A Discontinuous Galerkin Finite Element Method for Dynamic of Fully Saturated Soil / Rzwiazanie Zadania Dynamiki Całkowicie Nawodnionego Gruntu Przy Zastosowaniu Mes Z Nieciagłym Sformułowaniem Galerkina W Czasie

AbstractThe fully coupled, porous solid-fluid dynamic field equations with u-p formulation are used in this paper to simulate pore fluid and solid skeleton responses. The present formulation uses physical damping, which dissipates energy by velocity proportional damping. The proposed damping model takes into account the interaction of pore fluid and solid skeleton. The paper focuses on formulation and implementation of Time Discontinuous Galerkin (TDG) methods for soil dynamics in the case of fully saturated soil. This method uses both the displacements and velocities as basic unknowns and approximates them through piecewise linear functions which are continuous in space and discontinuous in time. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional central difference, Houbolt, Wilson θ, HHT- α, and Newmark methods. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than these traditional methods.

A nonlinear computational method for the propagation of shock waves in ducts: Application to buzz-saw noise.

A numerical method to compute shock wave propagation in uniform waveguides in 2-D and 3-D is presented. The solution is searched under the form of a modal solution of the Kuznetsov equation for which the modal amplitudes of the analytical linear modes are supposed to vary along the duct axis due to nonlinear interactions between the different modes and frequencies. This finally yields to a differential system on the mode amplitudes, which is solved numerically using a standard Runge-Kutta algorithm for ordinary differential equations after numerical truncation of the modal series. Examples are presented of the nonlinear evolution of the pressure field for a 2-D waveguide. One important 3-D application is the so-called “buzz-saw” noise occurring for high-bypass-ratio turbofan engines, when fan blade tip relative flows exceed the sound velocity. First results on 3-D simulations for nonlinear propagation of a saw-tooth waveform spiraling inside a hard-walled cylindrical will be presented. Extensions of the model to ducts with uniform flows or with liners (walls with impedance) will finally be discussed.

On estimation algorithms for ordinary differential equations

This article addresses the problem of estimating the parameters of a system of ordinary differential equations given data derived from noisy observations on the state variables. This problem is important in a range of applications in areas such as adaptive, real time control. There are two main classes of method for attacking this problem, and their equivalence and effectiveness (consistency) are discussed. Recent rate of convergence results for the major implementation techniques are summarized, and some matters requiring further consideration indicated. References H. G. Bock. Recent advances in parameter identification techniques in {ODE}. In P. Deuflhard and E. Hairer, editors, Numerical Treatment of Inverse Problems in Differential and Integral Equations , pages 95--121. Birkhauser, 1983. J. Nocedal and S. J. Wright. Numerical Optimization . Springer--Verlag, 1999. M. R. Osborne. Numerical questions in ODE boundary value problems. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006 , volume 48 of ANZIAM J. , pages C899--C926, February 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/79 [February 11, 2008]. M. R. Osborne. Fisher's method of scoring. Int. Stat. Rev. , 86:271--286, 1992. M. R. Osborne. The bock iteration for the ode estimation problem. 2008. in preparation. I. Tjoa and L. T. Biegler. Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic systems. Ind. Eng. Chem. Res. , 30:376--385, 1991.

Convergence rates for adaptive approximation of ordinary differential equations

This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by $\mathcal O(N)$ . The alternative version with constant tolerance stops with $\mathcal O(N\ {\rm log}\ N)$ total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the ${{L^{{\frac{{1}}{{p+1}}}}}}$ quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L 1 -norm of the error density.

LIMITATIONS OF DELAYED STATE FEEDBACK: A NUMERICAL STUDY

Stabilization of a class of linear time-delay systems can be achieved by a numerical procedure, called the continuous pole placement method [Michiels et al., 2000]. This method can be seen as an extension of the classical pole placement algorithm for ordinary differential equations to a class of delay differential equations. In [Michiels et al., 2000] it was applied to the stabilization of a linear time-invariant system with an input delay using static state feedback. In this paper we study the limitations of such delayed state feedback laws. More precisely we completely characterize the class of stabilizable plants in the 2D-case. For that purpose we make use of numerical continuation techniques. The use of delayed state feedback in various control applications and the effect of its limitations are briefly discussed.

The construction of order 4 DIMSIMs for ordinary differential equations

Diagonally Implicit Multistage Integration Methods (DIMSIMs) of type 1 and 2 have considerable potential as numerical algorithms for ordinary differential equations. The aim of this paper is to construct such methods of order 4 of type 1 and 2, which completes the set for orders 1–8.

A particular integral BEM/time-discontinuous FEM methodology for solving 2-D elastodynamic problems

This study proposes a time-discontinuous Galerkin finite element method (FEM) for solving second-order ordinary differential equations in the time domain. The equations are formulated using a particular integral boundary element method (BEM) in the space domain for elastodynamic problems. The particular integral BEM technique depends only on elastostatic displacement and traction fundamental solutions, without resorting to commonly used complex fundamental solutions for elastodynamic problems. Based on the time-discontinuous Galerkin FEM, the unknown displacements and velocities are approximated as piecewise linear functions in the time domain, and are permitted to be discontinuous at the discrete time levels. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional finite difference method (the Houbolt method). Both methods are employed for a particular integral BEM analysis in elastodynamics. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than the traditional finite difference methods.

Weighted Quasi-Newton and Variable-Order, Variable-Step Adams Algorithm for Determining Site-Specific Reaction Rate Constants

A weighted quasi-Newton algorithm for function minimization and a variable-order, variable-step Adams algorithm for ordinary differential equations were combined to solve site-specific gas-phase reaction rate constants. If the systems appear to be “stiff”, smaller steps were taken in Adams method to search for the solution. A user supplied error tolerance was used to determine the accuracy of the solution. Upon the return of each solution, a weighted algorithm is introduced into the calculation of Χ2 to minimize interference from noise. Lower and upper bounds for reagent fraction and rate constants were applied to ensure the validity of the results. The search direction is calculated by a quasi-Newton algorithm. When a saddle point is suspected, a local search is carried out with a view to moving away from the saddle point. The information obtained from site-specific rate constants may provide further insight into the reaction mechanism as well as gas-phase structure. It can be applied to gas-phase H/D ex...

Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

An algorithm for ODEs from atmospheric dispersion problems

The solution of large systems of ordinary differential equations (ODEs), arising from atmospheric dispersion problems is considered. An algorithm using a method due to Klopfenstein is adopted as the main method and combined with an approximate Jacobian factorization and a Gauss-Seidel iteration to provide an efficient solver. The approach is contrasted with that of using implicit-explicit multistep methods. Numerical experiments are presented to illustrate the performance of the method.

The use of Hamilton's principle to derive time-advance algorithms for ordinary differential equations

Hamilton's principle is applied to derive a class of numerical algorithms for systems of ordinary differential equations when the equations are derivable from a Lagrangian. This is an important extension into the time domain of an earlier use of Hamilton's principle to derive algorithms for the spatial operators in Maxwell's equations. In that work, given a set of expansion functions for spatial dependences, the Vlasov-Maxwell equations were replaced by a system of ordinary differential equations in time, but the question of solving the ordinary differential equations was not addressed. Advantageous properties of the new time-advance algorithms have been identified analytically and by numerical comparison with other methods, such as Runge-Kutta and symplectic algorithms. This approach to time advance can be extended to include partial differential equations and the Vlasov-Maxwell equations. An interesting issue that could be studied is whether a collisionless plasma simulation completely based on Hamilton's principle can be used to obtain a convergent computation of average properties, such as the electric energy, even when the underlying particle motion is characterized by sensitive dependence on initial conditions.

Integration algorithm for ordinary differential equations which produces analytical results

Problems of integration of systems of ordinary linear differential equations on large intervals are attracting increasing attention. Such problems include, in particular, problems of celestial mechanics, calculation of asymptotic expansions, and evaluation of special functions for large argument values. The existing methods do not always guarantee high accuracy on large intervals. It is therefore relevant to develop methods whose errors are nonincreasing with the increase of the distance from the initial point and which allow integration of systems of arbitrary order. The developments in computer applications, and primarily the larger memories and faster speeds of modern computers, make it possible to perform symbolic transformations on the data, which produce a solution of the Cauchy problem in formula form. For a wide class of differential equations, this ensures asymptotically nonincreasing errors and also allows some analytical investigations of the solution. Moreover, this provides an opportunity for supplementing the existing libraries and software packages with qualitatively new programs that approximate the solutions of a large class of differential equations [1, 21. Consider a system of N ordinary differential equations ax u) = A (t) × X (t), (1) dt

Multiprocessor implementation of algorithms for ordinary differential equations

Multiprocessor computers constitute a cost-effective alternative to supercomputers for applications that require the solution of large sets of ordinary differential equations. The efficient use of multiprocessor comput ers however, requires tools for generation of parallel applications. This paper presents a general methodology for parallelization of the integration algorithms through a heuristic procedure for static multiprocessor scheduling of the computations involved. The procedure partitions, distributes and schedules the entire computation, based on decomposition of the mathematical models. Both problems are addressed: (i) minimize the completion time for a specified number of processors, and (ii) minimize the number of processors required to achieve a specified performance. The procedure combines high performance and simplicity for a wide variety of mathematical models, numerical algorithms and multiprocessor architec tures.