Numerical Integration Of System Research Articles

On Hamel’s equations

This paper reviews recent results on the extension of Hame?s formalism to infinite-dimensional mechanical systems and to variational integrators. Of a particular interest are applications to the dynamics and numerical integration of systems with velocity constraints.

On phenomenon of scattering on resonances associated with discretisation of systems with fast rotating phase

Numerical integration of ODEs by standard numerical methods reduces a continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of system with one fast rotating phase lead to a situation of such kind: numerical solution demonstrate phenomenon of scattering on resonances that is absent in the original system.

Numerical integration of systems of delay differential-algebraic equations

The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.

A novel algorithm for the numerical integration of systems of ordinary differential equations arising in chemical problems

Simulation of large networks of chemical reactions via the numerical integration of large systems of ordinary differential equations is of growing importance in real-world problems. We propose an attractive novel numerical integration method, that is largely independent from ill-conditioning and is suitable for any nonlinear problem; moreover, the method, being exact for linear problems, is especially precise for quasi-linear problems, the most frequent kind in the real world. The method is based on a new approach to the computation of a matrix exponential, includes an automatic correction of rounding errors, is not too expensive computationally, and lends itself to a short and robust software implementation that can be easily inserted in large simulation packages. A preliminary numerical verification has been performed, with encouraging results, on two sample problems. The full source listing (in standard C language) of an academic version of the algorithm is freely available on request (e-mail address: Valerio.Parisi@roma2.infn.it), together with a very simple but very stiff chemical problem.

Determination of the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Medium-Thickness Shells

A method is developed to determine the axisymmetric geometrically nonlinear thermoelastoviscoplastic stress–strain state of branched laminated medium-thickness shells of revolution. The method is based on the hypotheses of a rectilinear element for the whole set of layers. The shells are subject to loads that cause a meridional stress state and torsion. They can consist of isotropic layers, which deform beyond the elastic limit, and elastic orthotropic layers. The relations of thermoviscoplastic theory, which describe simple processes of loading, are employed as the equations of state for the isotropic layers. The solution of the problem is reduced to numerical integration of systems of differential equations. The geometrically nonlinear elastoplastic state of a two-layer corrugated shell of medium thickness is calculated as an example

Analysis of the axisymmetric thermoelastoplastic state of branched laminated transversally isotropic shells

A technique is developed for determination of the axisymmetric thermoelastoplastic stress-strain state of branched laminated transversally isotropic shells of revolution under loads that cause the meridional stress state and torsion. The method is based on the rectilinear-element hypotheses for the whole package of layers. To describe the processes of active elastoplastic deformation of a transversally isotropic material, deformation-type equations, which are constructed without recourse to the plastic-potential existence condition, are used. The scalar functions in the constitutive equations depend on the shear-strain rate and temperature. The solution of the problem is reduced to numerical integration of systems of differential equations. An example of determination of the elastoplastic state of a two-layer cylindrical shell stiffened with a rigid ring support is presented.

Quasiperiodic solutions of variational problems of motion in a central force field

A method is proposed for computing nearly optimal trajectories of dynamic systems with a small parameter by splitting the original variational problem into two separate problems for "fast" and "slow" variables. The problem for "fast" variables is solved by improving the zeroth approximation — the extremals of the linearized problem — by the Ritz method. The solution of the problem for "slow" variables is constructed by passing from a discrete argument — the number of revolutions around the attracting center— to a continuous argument. The proposed method does not require numerical integration of systems of differential equations and produces a highly accurate approximate solution of the problem.

Method of approximating series for numerical integration of systems of differential equations

A method of numerical integration of systems of differential equations is proposed that can be used for equations that describe processes occurring in every field of physics, namely, fluid mechanics, nuclear physics, solid-state physics, etc. The APPROX program package, which implements the method of approximating series, makes it possible to write programs for computations in no more than 2–3 hours and reduces the calculation time by 1–2 orders in comparison with finite-difference methods.

ACCELERATED COMPUTATION OF VISCOUS INCOMPRESSIBLE FLOWS WITH HEAT TRANSFER

A new method for enhancing the convergence rates of iterative schemes for the numerical integration of systems of partial differential equations has been developed. It is termed the distributed minimal residual (DMR) method. The DMR method has been applied to incompressible Navier-Stokes equations with heat transfer. Alt numerical lest cases were obtained using explicit four-stage Runge-Kutla or Euler implicit lime integration. The DMR method was found to reduce compulation lime by 207ndash;60%, depending on the test case. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

Numerical Reservoir Simulation Using an Ordinary-Differential-Equations Integrator

Abstract The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models. Introduction The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications. SPEJ P. 255

Stability of the numerical integration of systems of ordinary differential equations arising in the application of the straight line method to some equations of oscillation theory

Stability of the numerical integration of systems of ordinary differential equations arising in the application of the straight line method to some equations of oscillation theory

Numerical integration of systems of differential equations of complex chemical reactions with any ratios of the rate constants

A method is proposed for numerical integration of systems of differential equations of complex chemical reactions with any ratios of the rate constants included in these equations. The method overcomes the difficulties due to the instability involved in conventional integration. For ethylene chlorination, calculations show that within the mechanism considered the solution of the system of kinetic equations is the most sensitive to changes in the rate constant of chlorine dissociation. The estimation of the effect of changes in the rate constants at different stages of the reaction by means of the proposed method is also discussed with methane oxidation as an example.

Détermination numérique des mouvements d'un coin salé

Salt wedges are apt to form in river estuaries under certain flow conditions and eventually to extend several kilometres upstream. Where the river bed is fairly even and the tide not strong enough for sudden flow reversals to take place, there is little exchange between the two layers and an interface can be assumed to exist. Slight exchange does in fact frequently take place, however, and the interface can be defined as the locus of points at which the fluid density is the mean of those of the two layers. The problem is thus one of unsteady flow of two fluids of different densities with a free surface and an interface. The equations assuming gradually varying flows lead to a differential system comprising equations (7), (8), (10), (11), (14) and (15). Equation (23) gives the slopes of the four characteristics in each section and relationship (30) is satisfied on each characteristic. If condition (28) is satisfied the four characteristics are real and the problem is hyperbolic. Equation (23) enables the characteristics of the internal wave system to be distinguished from those of the external wave system ; relationship (29) gives an approximate value for the internal wave system slopes c. The considered region comprises an upstream part without a salt layer and a downstream part with the two layers one above the other ; the salt wedge front progressing at the same speed as the lower layer V1 comes between the two layers. The problem is solved numerically for the normal point with the aid of a Lax-wendroff explicit finite difference scheme. The characteristics are used at the ends of the integration domain, but as only those originating inside the domain are used, boundary conditions are required. A discharge/time or stage/time relationship is assumed upstream, and downstream (at the river mouth) a real surface variation relationship. If a second downstream condition is required, a critical section is assumed to become established at the mouth of the river. The steady flow conditions obtained by numerical integration of system (36) are taken as the initial conditions. A computation programme has been developed for a study of salt wedge movements in the Grand Rhone below Arles.