Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept (Communications in Computational Physics, 2018)
We present an efficient and robust method for stress wave propagation problems(second order hyperbolic systems) having discontinuities directly in their secondorder form. Due to the numerical dispersion around discontinuities and lack of the inherentdissipation in hyperbolic systems, proper simulation of such problems are challenging.The proposed idea is to denoise spurious oscillations by a post-processingstage from solutions obtained from higher-order grid-based methods (e.g., high-ordercollocation or finite-difference schemes). The denoising is done so that the solutionsremain higher-order (here, second order) around discontinuities and are still free fromspurious oscillations. For this purpose, improved Tikhonov regularization approachis advised. This means to let data themselves select proper denoised solutions (sincethere is no pre-assumptions about regularized results). The improved approach can directlybe done on uniform or non-uniform sampled data in a way that the regularizedresults maintenance continuous derivatives up to some desired order. It is shown howto improve the smoothing method so that it remains conservative and has local estimatingfeature. To confirm effectiveness of the proposed approach, finally, some oneand two dimensional examples will be provided. It will be shown how both the numerical(artificial) dispersion and dissipation can be controlled around discontinuoussolutions and stochastic-like results.