Dispersion-corrected compact finite difference discretizations for the elastic Helmholtz equation in two and three dimensions
Abstract
We develop a family of compact, dispersion-corrected marker-and-cell (MAC)-based finite-difference discretizations for the elastic Helmholtz equation. While dispersion correction is well established for scalar Helmholtz problems, fewer approaches exist for the elastic problems, and most use non-compact stencils. To this end, we introduce a generalized MAC (GMAC) framework that enables compact discretization of the elastic Helmholtz equation. We prove that even though GMAC is non-symmetric, this discretization produces no numerical dissipation and no shear-mode separation. For certain weights, GMAC has lower polarization error than standard MAC. We also introduce an elastic dispersion correction (EDC) process, extending real-shifted-wavenumber dispersion correction methods from acoustic to elastic Helmholtz problems. We prove and demonstrate numerically that GMAC is second-order accurate. Furthermore, we show that EDC substantially reduces dispersion for standard MAC and can provide additional correction in some instances of GMAC. Dispersion analysis shows that the resulting method outperforms existing compact finite-difference discretizations in dispersion accuracy and is competitive with some non-compact schemes. We give numerical evidence for improved phase alignment in mildly heterogeneous media.
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BibTeX
@misc{yovel2026dispersioncorrected,
title = {{Dispersion-corrected compact finite difference discretizations for the elastic Helmholtz equation in two and three dimensions}},
author = {Rachel Yovel and Eli Turkel and Eran Treister},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05110},
eprint = {2610.05110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05110},
}
RIS
TY - UNPB
TI - Dispersion-corrected compact finite difference discretizations for the elastic Helmholtz equation in two and three dimensions
AU - Rachel Yovel
AU - Eli Turkel
AU - Eran Treister
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05110
UR - https://arxiv.org/abs/2610.05110
AB - We develop a family of compact, dispersion-corrected marker-and-cell (MAC)-based finite-difference discretizations for the elastic Helmholtz equation. While dispersion correction is well established for scalar Helmholtz problems, fewer approaches exist for the elastic problems, and most use non-compact stencils. To this end, we introduce a generalized MAC (GMAC) framework that enables compact discretization of the elastic Helmholtz equation. We prove that even though GMAC is non-symmetric, this discretization produces no numerical dissipation and no shear-mode separation. For certain weights, GMAC has lower polarization error than standard MAC. We also introduce an elastic dispersion correction (EDC) process, extending real-shifted-wavenumber dispersion correction methods from acoustic to elastic Helmholtz problems. We prove and demonstrate numerically that GMAC is second-order accurate. Furthermore, we show that EDC substantially reduces dispersion for standard MAC and can provide additional correction in some instances of GMAC. Dispersion analysis shows that the resulting method outperforms existing compact finite-difference discretizations in dispersion accuracy and is competitive with some non-compact schemes. We give numerical evidence for improved phase alignment in mildly heterogeneous media.
ER -