Numerical Solution Of Differential-algebraic Equations Research Articles

NUMERICAL SOLUTIONS OF DIFFERENTIAL-ALGEBRAIC EQUATIONS (DAEs) USING LEGENDRE POLYNOMIALS APPROXIMATIONS

In this paper, the Legendre polynomials approximations are studied for the numerical solutions of differential-algebraic equations (DAEs). An algorithm of the method constructed using Legendre polynomials approximations is developed to solve DAE systems. Two test problems are answered to illustrate the Legendre polynomials approximations, and then the obtained solutions are confronted with the exact solutions of the problems. As a first step, the power series of an allowed equation system is determined, in second step it is converted into Legendre polynomials approximations structure which allows an arbitrary order. Furthermore, to get numerical solution of DAEs with Legendre polynomials approximations, a Maple algorithm is implemented. Graphs of obtained solutions are sketched, and are created tables to compare the exact solutions by using Maple programming.

A fifth order implicit method for the numerical solution of differential-algebraic equations

An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.

The numerical solution of differential-algebraic equations by sinc-collocation method

In this article, numerical solution of differential-algebraic equations (DAEs), by means of the sinc-collocation method is considered. Properties of the sinc procedure are utilized to reduce the computation of the DAEs to systems of algebraic equations. It is well known that the sinc procedure converges to the solution at an exponential rate. To show the validity and efficiency of the present method, some examples are presented. The method is easy to implement and the results show that this method is very efficient, and can be applied to a large class of problems.

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.

On the Numerical Solution of Differential‐Algebraic Equations with Hessenberg Index‐3

Numerical solution of differential‐algebraic equations with Hessenberg index‐3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.

On the numerical solution of differential-algebraic equations with index-3

In this study, numerical solution of differential-algebraic equations (DAEs) with index-3 has been presented by Padé Approximation. An index reduction for differential-algebraic equation with index-3 is suggested. First index-3 system is reduced to index-1 system. Then we have applied the method for solving differential-algebraic equations with index-3 numerically.

The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs)

In this paper, the numerical solution of differential-algebraic equations is considered by Padé approximation method. We applied this method to an example which is a physical model problem. Firstly the physical model problem has been converted to power series, then the numerical solution of physical model problem was put into Padé series form. Thus we obtained numerical solution of physical model problem.

On the numerical solution of differential-algebraic equations by Padé series

In this paper, the solution for a differential-algebraic equation (DAE) can be expanded up to arbitrary order using MAPLE computer algebra systems (Frank, MAPLE V, CRC Press, Boca Raton, FL, 1996). First we calculate Power series of the given equations system then transform it into Padé series form, which gives an arbitrary order for solving DAE numerically.

C++ implementations of numerical methods for solving differential-algebraic equations

Object-oriented programming can produce improved implementations of complex numerical methods, but it can also introduce a performance penalty. Since computational simulation often requires intricate and highly efficient codes, the performance penalty of high-level techniques must always be weighed against the improvements they enable. These issues are addressed in a general object-oriented (OO) toolkit for the numerical solution of differential-algebraic equations (DAEs). The toolkit can be configured in several different ways to solve DAE initial-value problems with an adaptive multistep method. It contains a wrapped version of the Fortran 77 code DASPK and a translation of this to C++. Two C++ constructs for assembling the tools are provided, as are two implementations an important DAE test problem. Multiple configurations of the toolkit for DAE test problems are compared in order to assess the performance penalties of C++. The mathematical methods and implementation techniques are discussed in detail in order to provide heuristics for efficient OO scientific programming and to demonstrate the effectiveness of OO techniques in managing complexity and producing better code. The codes were tested on a variety of problems using publicly available Fortran 77 and C++ compilers. Extensive effficiency comparisons are presented in order to isolate computationally inefficient OO techniques. Techniques that caused difficulty in implementation and maintenance are also highlighted. The comparisons demonstrate that the majority of C++'s built-in support for OO programming has a negligible effect on performance, when used at sufficiently high levels, and provides flexible and extensible software for numerical methods.

A State-Space-Based Implicit Integration Algorithm for Differential-Algebraic Equations of Multibody Dynamics*

ABSTRACT An implicit numerical integration algorithm based on generalized coordinate partitioning is presented for the numerical solution of differential-algebraic equations of motion arising in multibody dynamics. The algorithm employs implicit numerical integration formulas to express independent generalized coordinates and their first time derivative as functions of independent accelerations at discrete integration times. The latter are determined as the solution of discretized equations obtained from state-space, second-order ordinary differential equations in the independent coordinates. All dependent variables in the formulation, including Lagrange multipliers, are determined by satisfying the full system of kinematic and kinetic equations of motion. The algorithm is illustrated using the implicit trapezoidal rule to integrate the constrained equations of motion for three stiff mechanical systems with different generalized coordinate dimensions. Results show that the algorithm is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

Numerical solution of differential-algebraic equations in mechanical systems simulation

The numerical solution of the differential-algebraic equations of motion of mechanical systems offers many computational challenges. In this paper we describe progress which has been made in understanding the formulation of the equations of motion from the viewpoint of numerical stability, and outline some of the difficulties which must be resolved for efficient and reliable numerical methods in real-time simulation of mechanical systems.

Numerical solution of differential-algebraic equations for constrained mechanical motion

The two most popular formulations of the equations of constrained mechanical motion, thedescriptor andstate-space forms, each have severe practical limitations. In this paper, we discuss and relate some proposed reformulations of the equations which have improved numerical properties.

Simultaneous Numerical Solution of Differential-Algebraic Equations

A unified method for handling the mixed differential and algebraic equations of the type that commonly occur in the transient analysis of large networks or in continuous system simulation is discussed. The first part of the paper is a brief review of existing techniques of handling initial value problems for stiff ordinary differential equations written in the standard form y' f(y, t) . In the second part one of these techniques is applied to the problem F(y, y', t)=0 . This may be either a differential or an algebraic equation as \partial F/ \partial y' is nonzero or zero. It will represent a mixed system when vectors F and y represent components of a system. The method lends itself to the use of sparse matrix techniques when the problem is sparse.