Two-step peer methods with continuous output

Schmitt, Bernhard A.; Weiner, Rüdiger; Beck, Steffen

doi:10.1007/s10543-012-0415-z

Cited by 12 publications

(7 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For methods with a lower triangular coefficient matrix G the s stage systems are decoupled. Such peer methods have been discussed in [2], [21]. Singly-implicit methods with constant main diagonal g ii = γ, i = 1, .…”

Section: Diagonally Implicit Peer Methodsmentioning

confidence: 99%

“…Before proceeding we note that there is the interesting class of FSAL methods (first same as last) [21] with a singular coefficient G. Hence, nonsingularity of G will be assumed only later on starting with subsection 3.3. The transformed matrix (3.2) is better suited for the analysis than (3.1) since the first diagonal block depends linearly on the two matrices Ŵ and Ẑ := G T ZG, but not on G alone.…”

Section: A Rank Revealing Congruence Transformationmentioning

confidence: 99%

“…= 1.80350113085004, and the nodes are (c i ) = (−0.889874593986289, 0.522100340305431, −0.297184898847891, 1). We do not go into more detail here since this method may not be of practical interest because efficient methods should satisfy several criteria like those listed in [21]. The construction of efficient A-stable peer methods will be the topic of a forthcoming paper.…”

Section: Apply a Final Congruence-similarity Transformation With The ...mentioning

confidence: 99%

See 2 more Smart Citations

Algebraic criteria for A-stability of peer two-step methods

Schmitt

2015

Preprint

Self Cite

View full text Add to dashboard Cite

A new criterion for A-stability of peer two-step methods is presented which is verifiable exactly in exact arithmetic by checking semi-definiteness of a certain test matrix. It depends on the existence of two positive definite weight matrices for a given method. Although the initial approach is different using properties of the numerical radius the criterion itself resembles the one from algebraic stability of General Linear Methods. Known numerical algorithms for the computation of the unknown weight matrices suffer from rank deficiencies of the test matrix. For sstage peer methods of order s − 1 this rank defect is identified with an explicit block diagonal decomposition of the test matrix in trivial and definite blocks. In the design of methods its coefficients are unknown and an explicit parametrization of A-stable peer methods of order s − 1 is presented with a weight matrix as parameter. This leads to a general existence result for any number of stages. The restrictions for efficient L-stable peer methods like diagonally-implicit and parallel ones are also discussed and such methods with 3 and 4 stages are constructed.

show abstract

Section: Diagonally Implicit Peer Methodsmentioning

confidence: 99%

Section: A Rank Revealing Congruence Transformationmentioning

confidence: 99%

Section: Apply a Final Congruence-similarity Transformation With The ...mentioning

confidence: 99%

See 1 more Smart Citation

Algebraic criteria for A-stability of peer two-step methods

Schmitt

2015

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…These methods have been introduced in [6]. In addition, four FSAL methods (first same as last: last stage of the previous time step is reused as the first stage of the current time step) with s = 3, 5, 7, 9, which have been introduced in [5], are included, which can use up to s − 1 cores.…”

Section: A Computational Structurementioning

confidence: 99%

“…Peer methods have been introduced by Schmitt and Weiner in 2004 [4]. The explicit methods included in EPPEER, which has been released in 2012, are described in [5], [6], [7]. These methods possess up to 8 independent stages, which can be computed in parallel on different cores.…”

Section: Introductionmentioning

confidence: 99%