A numerical comparison of Galerkin and Collocation Isogeometric approximations of acoustic wave problems

Abstract

We consider the Galerkin and Collocation Isogeometric approximations of the acoustic wave equation with absorbing boundary conditions, while the time discretization is based on second-order Newmark schemes. A numerical study investigates the properties of the two IGA methods as concerns stability thresholds, convergence errors, accuracy, computational time, and sparsity of the stiffness matrices varying the polynomial degree p, mesh size h, regularity k, and time step Δt. In order to compare the two IGA methods, we focus on two meaningful examples in the framework of wave propagation simulations: a test problem with an oscillatory exact solution having increasing wave number, and the propagation of two interfering Ricker wavelets. Numerical results show that the IGA Collocation method retains the convergence and stability properties of IGA Galerkin. In all examples considered, IGA Collocation is in general less accurate when we adopt the same choices of parameters p, k, h, and Δt. On the other hand, regarding the computational cost and the amount of memory required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the IGA Galerkin method, especially in the case of maximal regularity k=p−1 with increasing NURBS degree p.