Explicit computational wave propagation in micro-heterogeneous media

Abstract

Explicit time stepping schemes are popular for linear acoustic and elastic wave propagation due to their simple nature which does not require sophisticated solvers for the inversion of the stiffness matrices. However, explicit schemes are only stable if the time step size is bounded by the mesh size in space subject to the so-called CFL condition. In micro-heterogeneous media, this condition is typically prohibitively restrictive because spatial oscillations of the medium need to be resolved by the discretization in space. This paper presents a way to reduce the spatial complexity in such a setting and, hence, to enable a relaxation of the CFL condition. This is done using the Localized Orthogonal Decomposition method as a tool for numerical homogenization. A complete convergence analysis is presented with appropriate, weak regularity assumptions on the initial data.

Large‐scale simulation of shallow water waves via computation only on small staggered patches

“…The scheme employs a macroscale staggered grid of patches (Section 2.2) to ensure good wave properties. The scheme is a dynamic multiscale computational homogenization [19,20, e.g.]. An advantage of this approach is that a user does not have to do any preprocessing, neither computational as in numerical homogenization [18,19,37, e.g.…”

“…The scheme is a dynamic multiscale computational homogenization [19,20, e.g.]. An advantage of this approach is that a user does not have to do any preprocessing, neither computational as in numerical homogenization [18,19,37, e.g. ], nor algebraic as in classic homogenization [36, e.g.].…”

“…A major distinction with other work is that our scheme computes only on small well‐separated patches of space, whereas other multiscale approaches typically compute over all space: some examples are the multiscale finite element method [18, e.g. ], the numerical homogenization of Maier and Peterseim, 19 or the computational homogenization discussed by Geers et al 20 …”

Numerical upscaling for heterogeneous materials in fractured domains

“…The solution is integrated in time by the Crank-Nicolson method. We note that explicit time integrator such as the leap-frog method can also be used [1,14].…”

Reconstruction of Quasi-Local Numerical Effective Models from Low-Resolution Measurements

“…One such method is the Localized Orthogonal Decomposition (LOD) approach which provides effective models that provably cope with arbitrary rough coefficients in a large class of model problems including diffusion problems [15,17,27], elasticity [2,16] and wave propagation [1,10,11,26,37,41], without requiring periodicity or scale separation. This method allows us to explicitly characterize an operator G to prove Theorem 1 in Sect.…”