Abstract: This talk addresses the challenges of designing high-order implicit schemes for systems of hyperbolic conservation laws, particularly in the context of stiff problems where wave speeds span several orders of magnitude.

In such cases, explicit schemes necessitates small time-steps due to the stringent CFL stability criterion. Implicit time integration schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements, thereby bypassing the need for minute time-steps. This feature is particularly beneficial when tracking phenomena of interest which evolve on slower time scales than the fastest waves.
However, implicit methods are computationally expensive, as they require solving a system of equations, typically nonlinear, at each time-step. Achieving high-order accuracy adds another layer of complexity due to the need for nonlinear space limiters that prevent spurious oscillations, which further increase the nonlinearity of the problem.
The novel approach proposed here uses a first-order implicit scheme to pre-compute the nonlinearities introduced by the space-limiting procedure. This allows the high-order implicit scheme to remain nonlinear only due to the flux function, thereby reducing computational complexity. The method is investigated using DIRK (Diagonally Implicit Runge-Kutta) schemes for time integration, along with CWENO (Central Weighted Essentially Non-Oscillatory) or Discontinuous Galerkin reconstructions for space approximation. The coupling between time-integration and space-approximation is studied through approximate dissipation-dispersion relations.
We show that this approach enables accurate resolution of slow waves, even with large time-steps, offering a significant advantage in stiff regimes.