# Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

@article{Lucht1999IndexesAS, title={Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations}, author={W. Lucht and K. Strehmel and C. Eichler-Liebenow}, journal={BIT Numerical Mathematics}, year={1999}, volume={39}, pages={484-512} }

Linear partial differential algebraic equations (PDAEs) of the form Aut(t, x) + Buxx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B ∈ Rn×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions… Expand

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#### References

SHOWING 1-10 OF 18 REFERENCES

Discretization based indices for semilinear partial differential algebraic equations

- Mathematics
- 1998

Abstract Semilinear partial differential algebraic equations (PDAEs) of the form Au t ( t , x ) + Bu xx ( t , x ) + F ( u ) = f ( t , x ) ( F ( u ) ∈ R n is a nonlinear function of u ∈ R n ), where… Expand

The Index of an Infinite Dimensional Implicit System

- Mathematics
- 1999

The idea of the index of a differential algebraic equation (DAE) (or implicit differential equation) has played a fundamental role in both the analysis of DAEs and the development of numerical… Expand

The Numerical Solution Of Differential-Algebraic Systems By Runge-Kutta Methods

- Computer Science
- 1989

These lecture notes provide a self-contained and comprehensive treatment of the numerical solution of differential-algebraic systems using Runge-Kutta methods, and also extrapolation methods. Expand

ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS

- Mathematics
- 1984

In this paper we study the numerical solution of the differential/algebraic systems $F(t,y,y') = 0$. Many of these systems can be solved conveniently and economically using a range of ODE methods.… Expand

Differential/Algebraic Equations are not ODE's

- Mathematics
- 1982

This paper outlines a number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form ${\bf F}(t,{\bf y},{\bf y}') = {\bf 0}$.… Expand

Existence and accuracy for matrix refinement equations

- Mathematics
- 1996

A refinement equation is a functional equation of the form f(x) = Σ k=0 N c k f(2x - k). It is the starting point for the construction of wavelets and for subdivision schemes in approximation theory.… Expand

Numerical solution of initial-value problems in differential-algebraic equations

- Mathematics, Computer Science
- Classics in applied mathematics
- 1996

The DAE home page introduces theoretical advances Numerical analysis advancements DAE software DASSL Supplementary bibliography Index. Expand

A Sequential Regularization Method for Time-Dependent Incompressible Navier--Stokes Equations

- Mathematics
- 1997

The objective of the paper is to present a method, called the sequential regularization method (SRM), for the nonstationary incompressible Navier--Stokes equations from the viewpoint of… Expand

Differential and Integral Inequalities

- Mathematics
- 1970

In 1964 the author's mono graph "Differential- und Integral-Un gleichungen," with the subtitle "und ihre Anwendung bei Abschatzungs und Eindeutigkeitsproblemen" was published. The present volume grew… Expand

Ordinary Differential Equations

- Mathematics
- 1964

Foreword to the Classics Edition Preface to the First Edition Preface to the Second Edition Errata I: Preliminaries II: Existence III: Differential In qualities and Uniqueness IV: Linear Differential… Expand