Split generalized-α method: A linear-cost solver for multi-dimensional second-order hyperbolic systems

We propose a variational splitting technique for the generalized-a 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-alpha method for the temporal discretization while standard C-0 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 infinity 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 L-2 and H-1 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. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study introduces a variational splitting technique for the generalized-alpha method to solve hyperbolic partial differential equations. By using tensor-product meshes and conducting spectral analysis, the method's stability and efficiency are optimized.