Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such problems.An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations.At each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control subproblem.Because of its inevitably high dimensionality after space-time discretization,the parabolicoptimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required.It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations,and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm.To implement the ADMM efficiently,we propose an inexactness criterion that is independent of the mesh size of the involved discretization,and that can be performed automatically with no need to set empirically perceived constant accuracy a priori.The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly,yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by some numerical results.Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations,hyperbolic equations,convection-diffusion equations,and fractional parabolic equations.

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.