Hermite integrator for high-order mesh-free schemes

In most mesh-free methods, the calculation of interactions between sample points or "particles" is the most time-consuming. When we use mesh-free methods with high spatial orders, the order of the time integration should also be high. If we use usual Runge-Kutta schemes, we need to perform the interaction calculation multiple times per time step. One way to reduce the number of interaction calculations is to use Hermite schemes, which use the time derivatives of the right-hand side of differential equations, since Hermite schemes require a smaller number of interaction calculations than Runge- Kutta schemes do to achieve the same order. In this paper, we construct a Hermite scheme for a mesh-free method with high spatial orders. We performed several numerical tests with fourth-order Hermite schemes and Runge-Kutta schemes. We found that, for both Hermite and Runge-Kutta schemes, the overall error is determined by the error of spatial derivatives, for time steps smaller than the stability limit. The calculation cost at the time-step size of the stability limit is smaller for Hermite schemes. Therefore, we conclude that Hermite schemes are more efficient than Runge-Kutta schemes and thus useful for high-order mesh-free methods for Lagrangian hydrodynamics.