High-order explicit local time-stepping methods for damped wave equations

Abstract

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

Newmark local time stepping on high-performance computing architectures

“…Following the framework in [7,14,15], we divide the finite-element mesh into both fine and coarse element regions and correspondingly we split the DOFs as…”

“…As the element boundaries, and therefore the refinement-level boundaries, are explicitly coupled via the numerical flux, implementing LTS from the global DG formulation can be relatively straightforward. As noted in [15], the choice of spatial discretization does not impact the convergence or stability properties of the LTS method. Given our desire to use a SEM with its continuous nodal basis, we examine the terms B(I − P)u n and BPũ m to alter their structure to utilize R and F to make the coupling between coarse and fine explicit.…”

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

“…Subsequently, an efficient way to overcome this issue was implemented in several papers [50,81,82,56]. The idea was to separate the interfaces where the spatial and temporal steps are refined, and the same idea is used in the present paper.…”

Hybrid Seminumerical Simulation Scheme to Predict Transducer Outputs of Acoustic Microscopes

“…This is usually not fulfilled for acoustic microscopy or ultrasonic material testing, where the investigated object is immersed in a coupling fluid at several wavelength distance from the transducer [4], [26]. A possible relief to reduce discretizational efforts is achieved by applying higher order spatial approximations like p-FEM or hp-FEM [27], [28], higher order global approaches like the k-space methods [29], [30], and higher order adaptive time-stepping schemes [25], [31]. Despite a strong reduction of computational resources, those methods still have to deal with large acoustic domains (for the immersed case), which oftentimes are too large to be treated efficiently by FE methods.…”