# An implicit spatial and high-order temporal finite difference scheme for 2D acoustic modelling

Enjiang Wang^{1}

^{2}Yang Liu

^{1}

^{2}

^{3}

^{1} State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China.

^{2} CNPC Key Laboratory of Geophysical Prospecting, China University of Petroleum, Beijing 102249, China.

^{3} Corresponding author. Email: wliuyang@vip.sina.com

*Exploration Geophysics* - https://doi.org/10.1071/EG16094

Submitted: 27 July 2016 Accepted: 1 December 2016 Published online: 16 January 2017

## Abstract

The finite difference (FD) method exhibits great superiority over other numerical methods due to its easy implementation and small computational requirement. We propose an effective FD method, characterised by implicit spatial and high-order temporal schemes, to reduce both the temporal and spatial dispersions simultaneously. For the temporal derivative, apart from the conventional second-order FD approximation, a special rhombus FD scheme is included to reach high-order accuracy in time. Compared with the Lax-Wendroff FD scheme, this scheme can achieve nearly the same temporal accuracy but requires less floating-point operation times and thus less computational cost when the same operator length is adopted. For the spatial derivatives, we adopt the implicit FD scheme to improve the spatial accuracy. Apart from the existing Taylor series expansion-based FD coefficients, we derive the least square optimisation based implicit spatial FD coefficients. Dispersion analysis and modelling examples demonstrate that, our proposed method can effectively decrease both the temporal and spatial dispersions, thus can provide more accurate wavefields.

**Key words:** acoustic wave equation, high-order finite difference, implicit scheme, least squares, spatial dispersion, Taylor-series expansion, temporal dispersion.

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