Optimal order diagonally implicit Runge-Kutta methods
type: Item
instance of: scholarly article
title: Optimal order diagonally implicit Runge-Kutta methods
zbMATH Open document ID: 0795.65044
DOI: 10.1007/BF01990538
author: A. H. Al-Rabeh
published in: BIT
publication date: 1994-09-08T00:00:00Z
review text: The approximate numerical integration of the \(m\)-dimensional stiff initial value problem \(y'= f(x,y)\), \(y(x_ 0)= y_ 0\), using a \(q\)- stage diagonally implicit Runge-Kutta (DIRK) method: \(k^{(n)}_ i= f\Bigl(x_ n+ c_ i h,\;h\sum^ i_{j=1} a_{ij} k^{(n)}_ j\Bigr)\), \(i=1,\dots,q\) and \(y_{n+1}= y_ n+ h\sum^ q_{i=1} b_ i k^{(n)}_ i\), where there exists an \(s\), \(1\leq s\leq q\), such that \(a_{ss}\neq 0\), is discussed. It is shown that the order of a \(q\)-stage non-confluent DIRK method with non-zero weights cannot exceed \(q+1\). DIRK methods of orders 5 and 6 in 4 and 5 stages respectively are constructed. It is further shown that the optimal order of a non- confluent \(q\)-stage DIRK method with non-zero weights is \(q\), for \(q\geq 6\).
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 536863
zbMATH Keywords: optimal order
zbMATH Keywords: diagonally implicit Runge-Kutta method
zbMATH Keywords: stiff initial value problem
zbMATH Keywords: DIRK method
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 536863