Optimal order diagonally implicit Runge-Kutta methods
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 \(q\geq 6\).
scientific article; zbMATH DE number 536863
scientific article