We propose an algorithm for the solution of large-scale evolutionary equations (ODEs and discretized time-dependent PDEs), assuming that the solution and the right-hand side admit a compressed approximation via low rank matrices or tensors. The scheme begins with a collocation discretization in time, which turns a linear ODE into a linear system of algebraic equations. A low rank approximation to the solution of this system is computed via an alternating iteration, without solving the original large problem. This introduces a certain approximation error; however, we show that the conservation laws, defined by linear functions of the solution, can be preserved in the low rank scheme with the machine precision. One example of such functions is the probability normalization in the Fokker-Planck or chemical master equations. Numerical experiments demonstrate that the low rank normalization-preserving solution can be faster and more accurate than the stochastic simulation algorithms.