Local time-stepping and locally implicit schemes for linear acoustic wave
equations

Constantin Carle and Marlis Hochbruck, Karlsruhe Institute of Technology, Germany

For the spatially discretized linear acoustic wave equation, stability of explicit time integration schemes such as the leapfrog scheme can only be guaranteed under a CFL condition of the form $ \tau \lesssim h_{min} $, where $ \tau $ denotes the time-step size and $ h_{min} $ the diameter of the smallest element in the underlying mesh. In the case of locally refined meshes, where only a few mesh elements are small compared to the remaining coarse ones, this condition is the main bottleneck for the efficiency of explicit schemes. A similar situation occurs for heterogeneous material coefficients, which are large only in a small part of the computational domain.
To overcome this issue, we introduce local time-stepping (LTS) schemes which rely on the leapfrog scheme on the coarse part of the mesh. On the fine part we employ two variants: a stabilized leapfrog-Chebyshev scheme leading to an explicit LTS scheme and  $ \theta $ -schemes resulting in (locally) implicit schemes. For the space discretization we focus on a symmetric interior penalty discontinuous Galerkin discretization. After the construction of these schemes we sketch the stability and error analysis. Moreover, we present some numerical examples confirming the theoretical results.