A high‐order parallel Eulerian‐Lagrangian algorithm for advection‐diffusion problems on unstructured meshes

Abstract

SummaryIn this paper, we present a high‐order discontinuous Galerkin Eulerian‐Lagrangian method for the solution of advection‐diffusion problems onstaggered unstructured meshesin two and three space dimensions. The particle trajectories are tracked backward in time by means of ahigh‐orderrepresentation of the velocity field and a linear mapping from the physical to a reference system, hence obtaining a very simple and efficient strategy that permits to follow the Lagrangian trajectories throughout the computational domain. The use of an Eulerian‐Lagrangian discretization increases the overall computational efficiency of the scheme because it is the only explicit method for the discretization of convective terms that admits large time steps without imposing a Courant‐Friedrichs‐Lewy–type stability condition. This property is fully exploited in this work by relying on asemi‐implicitdiscretization of the incompressible Navier‐Stokes equations, in which the pressure is discretized implicitly; thus, the sound speed does not play any role in the restriction of the maximum admissible time step. The resulting mild Courant‐Friedrichs‐Lewy stability condition, which is based only on the fluid velocity, is here overcome by the adoption of the Eulerian‐Lagrangian method for the advection terms and an implicit scheme for the diffusive part of the governing equations. As a consequence, the novel algorithm is able to run simulation with a time step that is defined by the user, depending on the desired efficiency and time scale of the physical phenomena under consideration. Finally, a completeMessage Passing Interface parallelizationof the code is presented, showing that our approach can reach up to 96% of scaling efficiency.