An adaptive Tikhonov regularization is integrated with an h-adaptive grid-based scheme for simulation of elastodynamic problems, involving seismic sources with discontinuous solutions and random media. The Tikhonov method is adapted by a newly-proposed detector based on the MINMOD limiters and the grids are adapted by the multiresolution analysis (MRA) via interpolation wavelets. Hence, both small and large magnitude physical waves are preserved by the adaptive estimations on non-uniform grids. Due to developing of non-dissipative spurious oscillations, numerical stability is guaranteed by the Tikhonov regularization acting as a post-processor on irregular grids. To preserve waves of small magnitudes, an adaptive regularization is utilized: using of smaller amount of smoothing for small magnitude waves. This adaptive smoothing guarantees also solution stability without over smoothing phenomenon in stochastic media. Proper distinguishing between noise and small physical waves are challenging due to existence of spurious oscillations in numerical simulations. This identification is performed in this study by the MINMOD limiter based algorithm. Finally, efficiency of the proposed concept is verified by: 1) three benchmarks of one-dimensional (1-D) wave propagation problems; 2) P-SV point sources and rupturing line-source including a bounded fault zone with stochastic material properties.