Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index

Summary. We examine a class of symmetric collocation schemes for the solution of nonlinear boundary value problems for unstructured nonlinear systems of differential-algebraic equations with arbitrary index. We show that these schemes converge with the same orders as one would expect for ordinary differential equations. In particular, we show superconvergence for a special choice of the collocation points. We demonstrate the efficiency of the new approach with some numerical examples.Mathematics Subject Classification (2000): 65L10Revised version received November 21, 2003Supported by DFG research grant Ku964/4.Supported by DFG research grant Me790/11.     

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.     

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

In this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order

We consider systems of linear differential-algebraic equations of order ≥ 2 and present conditions under which there exist nonsingular transformations splitting the system into systems of linear ordinary differential equations of lower order and, possibly, an algebraic part.     

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.     

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (differential-algebraic equations) with properly stated leading term exhibiting a singularity of the first kind. The procedure is based on a modified defect correction principle, extending an established technique from the context of ordinary differential equations to the differential-algebraic case. Using recent convergence results for stiffly accurate collocation methods, we prove that the resulting error estimate is asymptotically correct. Numerical examples demonstrate the performance of this approach. To keep the presentation reasonably self-contained, some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis, are also briefly reviewed. The appendix contains a remark about the interrelation between collocation and implicit Runge-Kutta methods for the DAE case.     

Gauß-Manin determinants for rank 1 irregular connections on curves

Let be a smooth open curve over a field , where k is an algebraically closed field of characteristic 0. Let be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gau?-Manin connection . The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank. Received: 13 September 1999 / Published online: 17 August 2001     

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.     

On the stability of functionally fitted Runge–Kutta methods

Classical collocation RK methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted RK (FRK) methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the parameters of the problem, the time, the stepsize, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show how to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example and we provide further insight for separable methods with symmetric collocation points. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60     

Cohomologies of a double covering of a non-singular algebraic 3-fold

In this paper we study the Hodge numbers of a branched double covering of a smooth, complete algebraic threefold. The involution on the double covering gives a splitting of the Hodge groups into symmetric and skew-symmetric parts. Since the symmetric part is naturally isomorphic to the corresponding Hodge group of the base we study only the skew-symmetric parts and prove that in many cases it can be computed explicitly. Received: 6 March 2001 / in final form: 4 September 2001/ Published online: 4 April 2002     

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi‐CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red‐black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi‐CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001     

Legendre spectral method for the fractional Bratu problem

In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.     

Rational Chebyshev collocation method for the similarity solution of two dimensional stagnation point flow

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.     

Non-stiff integrators for differential–algebraic systems of index 2

Non-stiff differential-algebraic equations (DAEs) can be solved efficiently by partitioned methods that combine well-known non-stiff integrators from ODE theory with an implicit method to handle the algebraic part of the system. In the present paper we consider partitioned one-step and partitioned multi-step methods for index-2 DAEs in Hessenberg form and the application of these methods to constrained mechanical systems. The methods are presented from a unified point of view. The comparison of various classes of methods is completed by numerical tests for benchmark problems from the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

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)     

Improved linear multi-step methods for stochastic ordinary differential equations

We consider linear multi-step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise. We present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations. In previous work, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical-stochastic integrals. We show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile. Simulation results illustrate the theoretical findings.     

Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index-1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper. The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.