zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Convergence of Runge-Kutta methods applied to linear partial differential-algebraic equations. (English) Zbl 1072.65122

Linear partial differential-algebraic equations (PDAEs) of the form

Au t (t,x)+B(u xx (t,x)+ru x (t,x))+Cu(t,x)=f(t,x)

where t(t 0 ,t e ),x(-l,l) and A,B,C n,n are constant matrices, r,u,f[t 0 ,t e ]×[-l,l] n , and what is more interesting the matrix A is singular, what leads to the differential-algebraic problem.

In this case it is impossible to prescribe initial and boundary conditions for all components of the solution vector, they must fulfill certain conditions. A practical example how to do it is shown for the so called superconducting coil.

The problem of linear PDAEs is investigated from the numerical point of view. The discretization in space via finite differences is used and then a Runge-Kutta approximation of the method of lines for differential algebraic equations (MOL-DAE) is derived. When N is the number of space grid points, by a regular transformation the MOL-DAE of dimension nN is decoupled into N systems of dimension n and the Weierstrass-Kronecker transformation for each of these systems into a system of ordinary differential equations and an algebraic system. The so called differential time index of the PDAE is introduced which gives the Runge-Kutta approximation to these subsystems. The convergence of L-stable Runge-Kutta discretizations with constant step sizes is proved. The obtained order of convergence in time depends on the differential time index of the PDAE and on the boundary condition if they are homogeneous or not.

Included numerical examples confirm the theoretical results for the backward Euler method and 3-stage Radau IIA method.

65M12Stability and convergence of numerical methods (IVP of PDE)
65M20Method of lines (IVP of PDE)
65L06Multistep, Runge-Kutta, and extrapolation methods
35R10Partial functional-differential equations
35G15Boundary value problems for linear higher-order PDE