High order integrators obtained by linear combinations of symmetric-conjugate compositions

A new family of methods involving complex coefficients for the numerical integration differential equations is presented and analyzed. They are constructed as linear combinations symmetric-conjugate compositions obtained from a basic time-symmetric integrator order 2n (n≥1). The integrators 2(n+k), k=1,2,…, preserve time-symmetry up to 4n+3 when applied with real vector fields. If in addition system Hamiltonian scheme symplectic, then they also symplecticity 4n+3. We show that these well suited parallel implementation, thus improving their efficiency. Methods 10 based on 4th-order built tested comparison other standard procedures increase scheme.