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.     

非线性刚性微分-代数系统的波形松弛离散法

研究基于Runge-Kutta方法的波形松弛离散过程,得到新的刚性微分-代数系统的收敛理论,及该类系统解的存在性和惟一性,并用具体算例测试该理论的有效实用性.     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Waveform relaxation for fractional sub-diffusion equations

Numerical Algorithms - We report a new kind of waveform relaxation (WR) method for general semi-linear fractional sub-diffusion equations, and analyze the upper bound for the iteration errors. It...     

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.     

Waveform relaxation of partial differential equations

In this paper, we investigate a numerical method for the construction of an optimal set of quadrature rules in the sense of Borges (Numer. Math. 67, 271–288, 1994) for two or three definite integrals with the same integrand and interval of integration, but with different weight functions, related to an arbitrary multi-index. The presented method is illustrated by numerical examples.     

Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

     

Waveform relaxation method for stochastic differential equations with constant delay

This paper extends the waveform relaxation method to stochastic differential equations with constant delay terms, gives sufficient conditions for the mean square convergence of the method. A lot of attention is paid to the rate of convergence of the method. The conditions of the superlinear convergence for a special case, which bases on the special splitting functions, are given. The theory is applied to a one-dimensional model problem and checked against results obtained by numerical experiments.     

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.     

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.     

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.     

A convergence and existence result for differential-algebraic inclusions

A general class of multistep methods for the approximate solution of a differential-algebraic inclusion is considered. The convergence of these methods as well as the existence of a (generalized) solution of the inclusion is proved.     

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.     

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)     

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.     

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.