Split generalized-[formula omitted] method: A linear-cost solver for multi-dimensional second-order hyperbolic systems

Abstract

We propose a variational splitting technique for the generalized-α method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with the total number of degrees of freedom for multi-dimensional problems. We use the generalized-α method for the temporal discretization while standard C0 finite elements as well as isogeometric elements for spatial discretization. . We perform spectral analysis on the amplification matrix to establish the unconditional stability of the method and to show how splitting affects the overall behavior of the time marching scheme for finite time step sizes, including the standard stability analysis limits 0 and ∞ as particular cases. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In these examples, we compute the L2 and H1 norms of the error to show the optimal convergence of the discrete method in space and second-order accuracy in time. Lastly, we also use these tests to demonstrate the linear cost of the solver as the number of degrees of freedom grows.