A variable parameter embedded DIRK algorithm for the numerical integration of stiff systems of ODEs
The defining matrix of a single diagonally implicit Runge-Kutta (SDIRK) method has a single eigenvalue, \(\gamma\), of multiplicity s. This admits an efficient implementation as each step requires the solution of s systems of linear equations with the same matrix (I-h\(\gamma\) J) where J is a Jacobian matrix. For problems in which the change in J is slight, choosing a constant step h allows preservation of both the matrix and its LU-factorization. The author proposes that the step length may be allowed to change while maintaining \(h\gamma\) constant. This would tend to maintain efficiency in implementation. Unfortunately, constraints on \(\gamma\) to maintain order and A-stability precludes an unrestricted choice. Results from an (unclear) implementation on four standard problems suggest some improvement in efficiency is possible for some problems.
singly diagonally implicit Runge-Kutta method
scientific article; zbMATH DE number 4011817
scientific article