@ARTICLE{Łoś_Marcin_ADI-based,_2022, author={Łoś, Marcin and Behnoudfar, Pouria and Dobija, Mateusz and Paszynski, Maciej}, volume={70}, number={5}, journal={Bulletin of the Polish Academy of Sciences Technical Sciences}, pages={e141985}, howpublished={online}, year={2022}, abstract={The modeling of P-waves has essential applications in seismology. This is because the detection of the P-waves is the first warning sign of the incoming earthquake. Thus, P-wave detection is an important part of an earthquake monitoring system. In this paper, we introduce a linear computational cost simulator for three-dimensional simulations of P-waves. We also generalize our formulations and derivation for elastic wave propagation problems. We use the alternating direction method with isogeometric finite elements to simulate seismic P-wave and elastic propagation problems. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along the particular coordinate axis in the sub-steps. We show that the resulting problem matrix can be represented as a multiplication of three multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. The resulting system of linear equations can be factorized in linear O (N) computational cost in every time step of the semi-implicit method. We use our method to simulate P-wave and elastic wave propagation problems. We derive the condition for the stability for seismic waves; namely, we show that the method is stable when τ < C min\{ hx,hy,hz\}, where C is a constant that depends on the PDE problem and also on the degree of splines used for the spatial approximation. We conclude our presentation with numerical results for seismic P-wave and elastic wave propagation problems.}, type={Article}, title={ADI-based, conditionally stable schemes for seismic P-wave and elastic wave propagation problems}, URL={http://www.czasopisma.pan.pl/Content/124027/PDF/2962_BPASTS_2022_70_5.pdf}, doi={10.24425/bpasts.2022.141985}, }