A high order cell centred dual grid Lagrangian Godunov scheme

A first order cell centred Lagrangian Godunov scheme based upon the use of a dual grid to determine vertex velocities was presented by the author in [A.J. Barlow, P.L. Roe, A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput. Fluids, 46 (2011) 133–136]. A second order version of the scheme is presented and results obtained with the new scheme are compared against those obtained with a staggered grid compatible finite element scheme [A.J. Barlow, A compatible finite element multi-material ALE hydrodynamics algorithm, Int. J. Numer. Methods Fluids 56 (2008) 953–964]. The new scheme is shown to provide comparable shock capturing to the staggered grid method while retaining the benefits of reduced mesh imprinting, robustness and improved symmetry preservation observed for the first order cell centred scheme [A.J. Barlow, P.L. Roe, A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput. Fluids, 46 (2011) 133–136]. Two different approaches are also considered for moving the vertices using the dual grid approach, a method which reconstructs nodal velocities at the start of every timestep and a second that carries the nodal velocities as an additional variable.