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

Hassan Yousefi, Seyed Shahram Ghorashi, Timon Rabczuk

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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 languageEnglish
Pages (from-to)86-135
Number of pages50
JournalCommunications in Computational Physics
Issue number1
Publication statusPublished - 2016 Jun 22


  • Numerical (artificial) dispersion
  • Second order hyperbolic systems
  • Tikhonov regularization

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)


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