# 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}
}
• Published 1999
• Mathematics
• BIT Numerical Mathematics
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
Convergence of Runge--Kutta methods applied to linear partial differential-algebraic equations
• Mathematics
• 2005
We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form Aut(t,x) + B(uxx(t,x) + rux(t,x)) + Cu(t,x)=f(t,x), where A, B, C ∈ Rn,n and the matrix A is singular. WeExpand
A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method
• Mathematics
• 2013
We have considered linear partial differential algebraic equations (LPDAEs) of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and aExpand
Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations
• Mathematics
• 2019
We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknownExpand
Numerical Analysis of Nonlinear Partial Differential-Algebraic Equations: A Coupled and an Abstract Systems Approach
Various mathematical models in many application areas give rise to systems of partial differential equations and differential-algebraic equations (DAEs). These systems are called partial or abstractExpand
Index analysis and reduction of systems of quasi-linear partial-differential and algebraic equations
• Mathematics, Computer Science
• Comput. Chem. Eng.
• 2016
A new method is proposed for a systematic index reduction of quasi- linear PDAE systems to reveal quasi-linear combinations of the differential quantities in the high-index model which are invariant with respect to a specific independent variable. Expand
Index Concepts for Linear Mixed Systems of Differential-Algebraic and Hyperbolic-Type Equations
• Mathematics, Computer Science
• SIAM J. Sci. Comput.
• 2001
In order to classify these systems, some index notions that have already been introduced to treat parabolic-type problems are extended and it is shown that these indices may detect an artificial sensitivity with respect to perturbations, e.g., if the semidiscretization does not consider the information transport along the characteristics. Expand
A further index concept for linear PDAEs of hyperbolic type
For many technical systems the use of a refined network approach yields mathematical models given by initial-boundary value problems of partial differential algebraic equations (PDAEs). The boundaryExpand
Fully parallel methods for a class of linear partial differential-algebraic equations
This note deals with two fullyparallel methods for solving linear partial differential- algebraic equations (PDAEs) of the form: Aut + Bu = f(x,t) (1) where A is a singular, symmetric and nonnegativeExpand
On quasi-linear PDAEs with convection: applications, indices, numerical solution
• Mathematics
• 2002
For a class of partial differential algebraic equations (PDAEs) of quasi-linear type which include nonlinear terms of convection type, a possibility to determine a time and spatial index isExpand
Three-layer finite difference method for solving linear differential algebraic systems of partial differential equations
A boundary value problem is examined for a linear differential algebraic system of partial differential equations with a special structure of the associate matrix pencil. The use of an appropriateExpand

#### 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 ), whereExpand
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 numericalExpand
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
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
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 ofExpand
Differential and Integral Inequalities
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 grewExpand
Ordinary Differential Equations
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 DifferentialExpand