On Efficient Application of Implicit Runge-Kutta Methods to Large-Scale Systems of Index 1 Differential-Algebraic Equations

Abstract

In the paper we study how to integrate numerically large-scale systems of semi-explicit index 1 differential-algebraic equations by implicit Runge-Kutta methods. In this case, if Newton-type iterations are applied to the discrete problems we need to solve high dimension linear systems with sparse coefficient matrices. Therefore we develop an effective way for packing such matrices of coefficients and derive special Gaussian elimination for parallel factorization of nonzero blocks of the matrices. As a result, we produce a new efficient procedure to solve linear systems arising from application of Newton iterations to the discretizations of large-scale index 1 differential-algebraic equations obtained by implicit Runge-Kutta methods. Numerical tests support theoretical results of the paper.