High-Order Eulerian Incompressible Smoothed Particle Hydrodynamics with Transition to Lagrangian Free-Surface Motion

The incompressible Smoothed Particle Hydrodynamics (ISPH)
method is derived in Eulerian form with high-order smoothing kernels to
provide increased accuracy for a range of steady and transient internal
flows. Periodic transient flows, in particular, demonstrate high-order
convergence and accuracies approaching, for example, spectral mesh-based
methods. The improved accuracies are achieved through new high-order
Gaussian kernels applied over regular particle distributions with time
stepping formally up to 2nd order for transient flows. The approach can
be easily extended to model free surface flows by merging from Eulerian
to Lagrangian regions in an Arbitrary-Lagrangian-Eulerian (ALE) fashion,
and a demonstration with periodic wave propagation is presented. In the
long term, it is envisaged that the method will greatly increase the
accuracy and efficiency of SPH methods, while retaining the flexibility
of SPH in modelling free surface and multiphase flows.