Hybrid Eulerian-Lagrangian Semi-Implicit Time-Integrators

Hybrid Eulerian-Lagrangian semi-implicit time-integrators (HELSI) are presented which use the standard semi-implicit formulation as their starting point. The advantage of such an approach is that existing models which employ Eulerian semi-implicit time-integrators can easily be augmented to use the HELSI approach, that is, provided that either: a series of advection problems can be solved efficiently as in the operator-integration-factor splitting (OIFS) method or that the grid and underlying numerics allow the construction of efficient search and interpolation stencils typically required of semi-Lagrangian methods. Once the advection problem is solved the approach then follows the standard semi-implicit approach. An important caveat of this approach is that the semi-implicit time-integrator must be constructed around the implicit backward difference formulas.