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.
| Original language | English |
|---|---|
| Pages (from-to) | 86-135 |
| Number of pages | 50 |
| Journal | Communications in Computational Physics |
| Volume | 20 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2016 Jun 22 |
| Externally published | Yes |
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Keywords
- Numerical (artificial) dispersion
- Second order hyperbolic systems
- Tikhonov regularization
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)
Cite this
Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept. / Yousefi, Hassan; Ghorashi, Seyed Shahram; Rabczuk, Timon.
In: Communications in Computational Physics, Vol. 20, No. 1, 22.06.2016, p. 86-135.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept
AU - Yousefi, Hassan
AU - Ghorashi, Seyed Shahram
AU - Rabczuk, Timon
PY - 2016/6/22
Y1 - 2016/6/22
N2 - 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.
AB - 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.
KW - Numerical (artificial) dispersion
KW - Second order hyperbolic systems
KW - Tikhonov regularization
UR - http://www.scopus.com/inward/record.url?scp=84975867888&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84975867888&partnerID=8YFLogxK
U2 - 10.4208/cicp.101214.011015a
DO - 10.4208/cicp.101214.011015a
M3 - Article
AN - SCOPUS:84975867888
VL - 20
SP - 86
EP - 135
JO - Communications in Computational Physics
JF - Communications in Computational Physics
SN - 1815-2406
IS - 1
ER -