A comparison between Collocation and Galerkin Isogeometric approximation of acoustic wave problems.

Isogeometric collocation methods combine the high smoothness of NURBS basis functions with the low computational cost of collocation methods, generating sparser stiffness and mass matrices than the ones generated by isogeometric Galerkin methods. Our previous work investigated the approximation of 2D acoustic wave problems with proper absorbing boundary conditions by Galerkin IGA methods in space and Newmark’s explicit schemes in time (IGA-Gal-New). In this talk, we extend our study to IGA collocation explicit and implicit approximations (IGA-Col-New). A detailed numerical study on both Cartesian and NURBS domains illustrate the stability and convergence properties of the two Isogeometric Newmark methods with respect to the IGA and Newmark discretization’s parameters. The experimental results show that the stability thresholds of the methods depend linearly on h and inversely on p, confirming that the proposed IGA-Col-New method retains the good convergence and stability properties of standard IGA-Gal-New and Spectral Element discretizations of acoustic problems. Moreover, a detailed comparison of convergence errors, CPU time, and matrix sparsity patterns show that IGA-Col-New often outperforms IGA-Gal-New, in particular in the case of maximal regularity k = p - 1 and for increasing NURBS degree p. Some numerical results on the spectral properties of the IGA-Col-New matrices are also mentioned.