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Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

Part of: Partial differential equations, boundary value problems Hyperbolic equations and systems

Published online by Cambridge University Press:  22 June 2016

Hassan Yousefi ,
Seyed Shahram Ghorashi  and
Timon Rabczuk
Hassan Yousefi*
Affiliation:
Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423 Weimar, Germany
Seyed Shahram Ghorashi*
Affiliation:
Research Training Group 1462, Bauhaus-Universität Weimar, 99423 Weimar, Germany
Timon Rabczuk*
Affiliation:
Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
*
*Corresponding author. Email addresses:hassan.yousefi@uni-weimar.de; hyosefi@ut.ac.ir (H. Yousefi), shahram.ghorashi@uni-weimar.de (S. Sh. Ghorashi), timon.rabczuk@tdt.edu.vn (T. Rabczuk)
*Corresponding author. Email addresses:hassan.yousefi@uni-weimar.de; hyosefi@ut.ac.ir (H. Yousefi), shahram.ghorashi@uni-weimar.de (S. Sh. Ghorashi), timon.rabczuk@tdt.edu.vn (T. Rabczuk)
*Corresponding author. Email addresses:hassan.yousefi@uni-weimar.de; hyosefi@ut.ac.ir (H. Yousefi), shahram.ghorashi@uni-weimar.de (S. Sh. Ghorashi), timon.rabczuk@tdt.edu.vn (T. Rabczuk)
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Abstract

We present an efficient and robust method for stress wave propagation problems (second order hyperbolic systems) having discontinuities directly in their second order form. Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems, proper simulation of such problems are challenging. The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods (e.g., high-order collocation or finite-difference schemes). The denoising is done so that the solutions remain higher-order (here, second order) around discontinuities and are still free from spurious oscillations. For this purpose, improved Tikhonov regularization approach is advised. This means to let data themselves select proper denoised solutions (since there is no pre-assumptions about regularized results). The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order. It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature. To confirm effectiveness of the proposed approach, finally, some one and two dimensional examples will be provided. It will be shown how both the numerical (artificial) dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.

Keywords

Second order hyperbolic systemsTikhonov regularizationNumerical (artificial) dispersion

MSC classification

Secondary: 35L51: Second-order hyperbolic systems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 86 - 135
Copyright
Copyright © Global-Science Press 2016 

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