In the paper we consider index~1 differential-algebraic systems of the form $$ x'(t)=gbigl(x(t),y(t)bigr),quad y(t)=fbigl(x(t),y(t)bigr), eqno(1) $$ where $t in [0,T]$, $x(t) in mbox {bf R}^m$, $y(t) in mbox {bf R}^n$, $g:D subset mbox {bf R}^{m+n} to {bf R}^m$, $f:D subset mbox {bf R}^{m+n} to {bf R}^n$, and the initial conditions $x(0)=x^0, y(0)=y^0$ are consistent; i.~e., $y^0=f(x^0,y^0)$. To solve problem (1) numerically, we apply an $l$-stage implicit Runge-Kutta (RK) and obtain the discrete system. Further, we apply Newton-type iterations due to their universality, however, the usage of such iterations leads us to the necessity to solve discrete systems of dimension~$(m+n)l$ many times during the integration. The latter allows us to conclude that Newton-type iterations put severe requirements on RAM and CPU time caused by the increasing dimension of the discrete problem in $l$ times when an $l$-stage implicit RK method has been used. To meet these requirements and to decrease the above parameters significantly we develop the idea of parallel factorization of nonzero blocks of sparse coefficient matrices of the linear systems arising from discretization of index~1 differential-algebraic problems by RK methods and their following solving by Newton-type iterations. We formulate a number of theorems that give estimates for the local fill-in of such matrices on some stages of Gaussian elimination. As the result, we derive that only the suggested modification of Gauss method appeared to be effective and economical one from the standpoint of CPU time and RAM. Finally, we come to the concept of vector Gauss method which is the best one to implement high order implicit RK methods in practice.

CEMAT - Center for Computational and Stochastic Mathematics