On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.     

Regularization of differential-algebraic equations

Linear systems of ordinary differential equations with an identically singular or rectangular matrix multiplying the derivative of the unknown vector function are numerically solved by applying the least squares method and Tikhonov regularization. The deviation of the solution of the regularized problem from the solution set of the original problem is estimated depending on the regularization parameter.     

The Lagrange differential-algebraic equations

The Lagrange equations are considered in the case when the Lagrangen function is independent of some of the velocities, and the properties of the differential-algebraic equations arising are studied. It is proved that, when non-degeneracy conditions are satisfied, the equations arising reduce to differential Lagrange equations of lower dimension. The problem of the planar oscillations of an elastic pendulum is considered as an example.     

On singular points of quasilinear differential and differential-algebraic equations

We show how certain singularities of quasilinear differential and differential-algberaic equations can be resolved by taking the solutions to be integral manifolds of certain distributions rather than curves with specific parametrization.     

On a least-squares collocation method for linear differential-algebraic equations

Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.     

On qualitative properties of the solutions of semilinear elliptic equations

Semilinear elliptic equations of an arbitrary order 2m are considered. A theorem on the removable singularities of the solutions and a Liouville type theorem are proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 186–190, 1992.     

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

For linear constant-coefficient differential-algebraic equations, we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval. In particular, we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions, which may have wider and more useful application extent. Numerical simulations demonstrate the validity of the theory.     

Boundary value problems for differential-algebraic equations

This paper shows how certain regularization methods can be used to determine the solution structure of linear differential systems subject to linear constraints and boundary conditions. Attention is restricted to the relatively straightforward index-one problems and to certain index-two problems, where endpoint jumps are related to the underlying boundary layer structure of the corresponding singular perturbation problems.     

Defect-based a-posteriori error estimation for differential-algebraic equations

We show how the QDeC estimator, an efficient and asymptotically correct a-posteriori error estimator for collocation solutions to ODE systems, can be extended to differential-algebraic equations of index 1. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.     

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.     

Nonnegativity of solutions of nonlinear fractional differential-algebraic equations

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.     

Collocation methods for differential-algebraic equations of index 3

Summary This article give sharp convergence results for stiffly accurate collocation methods as applied to differential-algebraic equations (DAE's) of index 3 in Hessenberg form, proving partially a conjecture of Hairer, Lubich, and Roche.     

Adomian decomposition method for solution of differential-algebraic equations

Solutions of differential algebraic equations is considered by Adomian decomposition method. In E. Babolian, M.M. Hosseini [Reducing index and spectral methods for differential-algebraic equations, J. Appl. Math. Comput. 140 (2003) 77] and M.M. Hosseini [An index reduction method for linear Hessenberg systems, J. Appl. Math. Comput., in press], an efficient technique to reduce index of semi-explicit differential algebraic equations has been presented. In this paper, Adomian decomposition method is applied to reduced index problems. The scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods.     

On a differential-algebraic problem

The method of quasilinearization is a procedure for obtaining approximate solutions of differential equations. In this paper, this technique is applied to a differential-algebraic problem. Under some natural assumptions, monotone sequences converge quadratically to a unique solution of our problem.     

On first- and second-order difference schemes for differential-algebraic equations of index at most two

Difference schemes of the Euler and trapezoidal types for the numerical solution of the initial-value problem for linear differential-algebraic equations are examined. These schemes are analyzed for model examples, and their superiority over the familiar first- and second-order implicit methods is shown. Conditions for the convergence of the proposed algorithms are formulated.     

Passivity-preserving model reduction of differential-algebraic equations in circuit simulation

We present an extension of the positive real balanced truncation model reduction method for differential-algebraic equations that arise in circuit simulation. This method is based on balancing the solutions of the projected generalized algebraic Riccati equations. Important properties of this method are that passivity is preserved in the reduced-order model and that there exists an approximation error bound. Numerical solution of the projected Riccati equations using the special structure of circuit equations is also discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelet-based adaptive grids for multirate partial differential-algebraic equations

The mathematical model of electric circuits yields systems of differential-algebraic equations (DAEs). In radio frequency applications, a multivariate model of oscillatory signals transforms the DAEs into a system of multirate partial differential-algebraic equations (MPDAEs). Considering quasiperiodic signals, an approach based on a method of characteristics yields efficient numerical schemes for the MPDAEs in time domain. If additionally digital signal structures occur, an adaptive grid is required to achieve the efficiency of the technique. We present a strategy applying a wavelet transformation to construct a mesh for resolving steep gradients in respective signals. Consequently, we employ finite difference methods to determine an approximative solution of characteristic systems in according grid points. Numerical simulations demonstrate the performance of the adaptive grid generation, where radio frequency signals with digital structures are resolved.