Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

Abstract

Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity. However, 3D complex geologic settings often contain details on a very small scale compared to the dominant wavelength, requiring the mesh to contain elements that are smaller than dictated by the wavelength. Also, limitations of the mesh generation software may produce regions where the elements are much smaller than desired. In both cases, this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost. Local time stepping (LTS) can improve the computational efficiency and speed up the simulation. We evaluated a local formulation of an LTS scheme with second-order accuracy for the discontinuous Galerkin finite-element discretization of the wave equation. We tested the benefits of the scheme by considering a geologic model for a North-Sea-type example.

Newmark local time stepping on high-performance computing architectures

“…The previously defined algorithm does not make explicit considerations for an efficient implementation with continuous finite elements that can achieve the LTS speedup predicted by (28).…”

“…Further work using a DG method done by Gödel et al [13] was able to show an LTS algorithm working on GPUs for Maxwell's equations. The LTS-leap-frog method proposed by [7] was implemented by [28] using a DG discretization for applications in seismic wave propagation. We note that all of these successful, high-performance implementations of LTS for wave propagation applications have utilized a DG discretization, which may not always be desired.…”

A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters

“…The DG approach does not suffer from these limitations, but it is generally considered as expensive because of the excessively large number of degrees of freedom. In contrast with other methods, DG schemes are easily combined with local time-stepping strategies in order to improve computational efficiency and speed up of geophysical wave propagation [Baldassari et al, 2011, Dumbser and Käser, 2009, Minisini et al, 2013.…”

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

“…Here, we summarize methods of direct time integration suitable for usage in dynamic FE computations [13]: explicit methods [2,3,5,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], implicit methods [32][33][34][35][36][37][38], implicit-explicit methods [39][40][41], multi-time step and time sub-cycling methods [31,42,43], heterogeneous and asynchronous time integrators [44][45][46], variational time integrators [47,48], various local stepping approaches in time [49][50][51][52][53], time schemes for higher-order FEM and isogeometric analysis [2,54], and methods based on binary partitioning using a variable time step size…”