On numerical solution of large-scale systems of index 1 differential-algebraic equations

In this 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 we need to solve high dimension linear systems with sparse coefficient matrices. We develop an effective way for packing such matrices of coefficients. We also derive a special Gaussian elimination for parallel factorization of nonzero blocks of the matrix. As a result, we produce a new efficient procedure to solve linear systems arising in an application of implicit Runge–Kutta methods to large-scale differential-algebraic equations of index 1. Numerical examples support theoretical results of the paper.