Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems

An enhanced interpolation wavelet-based adaptive-grid scheme is implemented for simulating high gradient smooth solutions (as well as, discontinuous ones) in elastodynamic problems in domains with irregular boundary shapes. In the method, spatially adaptive smoothing is used to improve interpolation property of the solution in high gradient zones. In hyperbolic systems, in fact, there are no certain inherent regularities; hence, the erroneous adapted grid may be achieved because of small spurious oscillations in the solution domain. These oscillations, mainly formed in the vicinity of high gradient and discontinuity zones, make the adaptation procedure strongly unstable. To cover this drawback, enhanced smoothing splines are used to denoise directly non-physical oscillations in the irregular grid points, a kind of ill-posed problem. Controllable smoothing is achieved using non-uniform weight coefficients. As the smoothing splines are a kind of the Thikhonov regularization method, they work stably in irregular grid points. Regarding the Thikhonov regularization method, L-curve scheme could be used to investigate trade-off between accuracy and smoothness of the solutions. This relationship, in fact, could not be reliably captured by common computational methods. The proposed method, in general, is easy and conceptually straightforward; as all calculations are carried out in the physical domain. This concept is verified using a variety of 2D numerical examples.