Invited Lecture
On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs
University of Ferrara, Italy
Abstract
This talk is devoted to present the construction of a new class of semi-Lagrangian (SL) schemes with
implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The
semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK
strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is
considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian
IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while
an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable
numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum
admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with
multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative
scheme is then crucial to properly capture the wave propagation speed and thus to locate the
discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based
on the integration of the governing equations over the space-time control volume which arises from the
motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined
with a Cauchy-Kowalevskaya procedure that provides a predictor solution within each space-time
element. The one-dimensional shallow water equations (SWE) are chosen to validate the new
conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly,
respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a
relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative
and the conservative version of the novel class of methods demonstrates that the formal order of accuracy
is achieved and numerical evidences about the conservation property are shown. The AP property for the
corresponding relaxation system is also investigated.
References
[1] W. Boscheri, M. Tavelli, L. Pareschi. On the construction of conservative semi-Lagrangian IMEX
advection schemes for multiscale time dependent PDEs. Journal of Scientific Computing, accepted (2022).