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In this paper, a swarm trajectory-planning method is proposed for multiple autonomous surface vehicles (ASVs) in an unknown and obstacle-rich environment. Specifically, based on the point cloud information of the surrounding environment obtained from local sensors, a kinodynamic path-searching method is used to generate a series of waypoints in the discretized control space at first. Next, after fitting B-spline curves to the obtained waypoints, a nonlinear optimization problem is formulated to optimize the B-spline curves based on gradient-based local planning. Finally, a numerical optimization method is used to solve the optimization problems in real time to obtain collision-free, smooth and dynamically feasible trajectories relying on a shared network. The simulation results demonstrate the effectiveness and efficiency of the proposed swarm trajectory-planning method for a network of ASVs.