Modification of the spectral method for the description of shock wave propagation

A new semi-analytical modification of a frequency-domain numerical approach is presented for the description of nonlinear waves containing shocks. A known high-frequency asymptotic behavior of the spectrum of a wave with a singularity as a shock is taken into account in order to reduce the number of spectral components for the accurate numerical solution of nonlinear problem. A finite set of coupled equations for spectral amplitudes which approximates the infinite set of equations corresponding to an exact spectral formulation of the Burgers type equation is presented. High-frequency components are approximated by analytic asymptote of the sawtooth-like wave, that provides a main term in high frequency series expansion of the entire wave spectrum. Several model problems are considered to verify the accuracy and stability of the numerical scheme. It is shown that in comparison with direct spectral algorithm, this approach provides an adequate description of the shocked nonlinear waves and pulses with much less (about 10 times) number of harmonics