@Article{CiCP-20-86,
author = {Hassan Yousefi, Seyed Shahram Ghorashi and Timon Rabczuk},
title = {Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept},
journal = {Communications in Computational Physics},
year = {2018},
volume = {20},
number = {1},
pages = {86--135},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.101214.011015a},
url = {http://global-sci.org/intro/article_detail/cicp/11146.html}
}