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Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

Published online by Cambridge University Press:  03 June 2015

N. Anders Petersson  and
Björn Sjögreen
N. Anders Petersson*
Affiliation:
Center for Applied Scientific Computing, L-422, LLNL, P.O. Box 808, Livermore, CA 94551, USA
Björn Sjögreen*
Affiliation:
Center for Applied Scientific Computing, L-422, LLNL, P.O. Box 808, Livermore, CA 94551, USA
*
Corresponding author.Email:petersson1@llnl.gov
Email:sjogreen2@llnl.gov
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Abstract

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

Keywords

65M0674S2086A15Far field boundary conditionshigher order methodselastic wave equationsummation by parts
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 913 - 955
Copyright
Copyright © Global Science Press Limited 2014

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