In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlin...In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.展开更多
We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelera...We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent(PAGD)method.We discretize these problems with a Fourier collocation method in space,and employ various second-order schemes in time.We observe a significant speedup with this solver when compared to the preconditioned gradient descent(PGD)method.With the PAGD solver,fully implicit,second-order-in-time schemes are not only feasible to solve the PFC and FCH equations,but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account.Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that,for the FCH experiments,the fully implicit schemes(midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver)perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision.For the PFC,the results are not as conclusive as in the FCH experiments,which,we believe,is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation.We also discuss some practical matters in applying the PAGD.We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters.The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.展开更多
基金The research of the first author is supported by the Hong Kong Baptist University. The research of the second author is partially supported by a USA-AR0 grant 43751-MA and USA- NFS grants DMS0201094 and DMS-0412654. The third author is partially supported by CERG Grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, and an NSAF Grant （#10476032） of National Science Foundation of Chian.
文摘In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.
基金NSF grants DMS-1720213,DMS-1719854,and DMS-2012634NSF grants DMS-1720213 and DMS-2111228.The work of S.M.Wise was partially supported by DMS-1719854 and DMS-2012634.
文摘We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent(PAGD)method.We discretize these problems with a Fourier collocation method in space,and employ various second-order schemes in time.We observe a significant speedup with this solver when compared to the preconditioned gradient descent(PGD)method.With the PAGD solver,fully implicit,second-order-in-time schemes are not only feasible to solve the PFC and FCH equations,but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account.Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that,for the FCH experiments,the fully implicit schemes(midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver)perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision.For the PFC,the results are not as conclusive as in the FCH experiments,which,we believe,is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation.We also discuss some practical matters in applying the PAGD.We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters.The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.
