Abstract
In this paper, we study in detail dispersive properties of the widely used implicit scheme called Newmark trapezoidal rule in conjunction with modified mass matrix to enjoy superior dispersive properties keeping the finite element stencil compact. We call such schemes compact modified implicit finite element scheme. In case of one-dimensional propagation, following contributions are made: (a) for modified finite element (MFE) scheme, we find an optimal value of dispersion controlling parameter depending on Courant-Friedrichs-Lewy (CFL) number which provides exact solutions at the nodes of spatial gird; (b) for standard finite element (SFE) scheme optimal value of CFL number is obtained which provides fourth-order accurate solutions. Moreover, in case of two-dimensional propagation following contributions are made: (c) we have found optimal value of CFL number for all angles in case of both SFE and MFE schemes; (d) superior dispersive behaviour is evident in case of MFE scheme in comparison with SFE scheme. Furthermore, the MFE scheme can be efficiently implemented using non-standard quadrature rules or just updating mass matrix which does not require to write brand new code and makes it computationally very attractive. Also for specific value of parameter, i.e. tau = 0, the MFE scheme leads back to the SFE scheme.