New numerical integrators based on solvability and splitting

When Lie-group integrators such as those based on the Magnus expansion are applied to linear systems of ODEs, it is necessary to evaluate matrix exponentials. This leads to a reduction in their computational efficiency when the dimension of the matrix is very large. For quadratic Lie groups it is possible to approximate the matrix exponential by a rational function and still preserve the Lie-group structure, but this is no longer true in the important case of the special linear group. In this paper we propose a new class of integration algorithms especially designed to avoid this problem. They are based on expressing the solution as a product of upper and lower triangular matrices obtained explicitly in terms of quadratures. We analyse the main features of the procedure and its feasibility as a practical numerical method.