Before revision: Solving wave equations in a time-parallel manner is challenging, and the algorithm based on the block \alpha-circulant preconditioning technique has shown its advantage in many existing studies (where \alpha\in(0, 1) is a free parameter).
After revision: Solving wave equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block \alpha-circulant preconditioning technique has shown its advantage in many existing studies (where \alpha\in(0, 1) is a free parameter).