Error estimates and parallel evaluation of hybrid schemes for parabolic, wave, and Schrödinger equations

In this paper, we study error estimates of parallel evaluation for two types of hybrid difference schemes: HIEuler scheme and HBDF2 scheme. Each scheme is composed of the explicit midpoint scheme at the intermediary time-steps combined with the implicit Euler method/the backward difference formula at the final time-step. The key ingredient lies in that error estimates are rigorously proved under the parallel setting with the help of the energy method. To reduce storage requirements and computational costs, efficient parallel solvers for the parabolic, wave and Schrödinger equations are developed, respectively. Finally, several numerical examples are carried out to verify theoretical findings.