Abstract:
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equation for neutral and ionic solutions, respectively. In the present work solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented to the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of a ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency, and allow for the treatment of different boundary conditions, as for example surface systems. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.

Abstract:
A fast Poisson solver software package PoisFFT is presented. It is available as a free software licensed under the GNU GPL license version 3. The package uses the fast Fourier transform to directly solve the Poisson equation on a uniform orthogonal grid. It can solve the pseudo-spectral approximation and the second order finite difference approximation of the continuous solution. The paper reviews the mathematical methods for the fast Poisson solver and discusses the software implementation and parallelization. The use of PoisFFT in an incompressible flow solver is also demonstrated.

Abstract:
We discuss the scalable parallel solution of the Poisson equation within a Particle-In-Cell (PIC) code for the simulation of electron beams in particle accelerators of irregular shape. The problem is discretized by Finite Differences. Depending on the treatment of the Dirichlet boundary the resulting system of equations is symmetric or `mildly' nonsymmetric positive definite. In all cases, the system is solved by the preconditioned conjugate gradient algorithm with smoothed aggregation (SA) based algebraic multigrid (AMG) preconditioning. We investigate variants of the implementation of SA-AMG that lead to considerable improvements in the execution times. We demonstrate good scalability of the solver on distributed memory parallel processor with up to 2048 processors. We also compare our SAAMG-PCG solver with an FFT-based solver that is more commonly used for applications in beam dynamics.

Abstract:
A method is described to solve the Poisson problem for a three dimensional source distribution that is periodic into one direction. Perpendicular to the direction of periodicity a free space (or open) boundary is realized. In beam physics, this approach allows to calculate the space charge field of a continualized charged particle distribution with periodic pattern. The method is based on a particle mesh approach with equidistant grid and fast convolution with a Green's function. The periodic approach uses only one period of the source distribution, but a periodic extension of the Green's function. The approach is numerically efficient and allows the investigation of periodic- and pseudo-periodic structures with period lengths that are small compared to the source dimensions, for instance of laser modulated beams or of the evolution of micro bunch structures. Applications for laser modulated beams are given.

Abstract:
We describe an implementation to solve Poisson's equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory parallel computer. Test results to demonstrate the convergence and scaling properties of the implementation are presented. The solver is offered to interested users as the library PSPFFT.

Abstract:
We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kind of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.

Abstract:
A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Furthermore, some results are presented for a charged colloidal sphere surrounded by counterions.

Abstract:
The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of $~\boldsymbol{k} \boldsymbol{M}+ \boldsymbol{M} \boldsymbol{k} = \boldsymbol{r} {\boldsymbol{c}}^{T}-g\boldsymbol{K}^{-1} \boldsymbol{r} {\boldsymbol{c}}^{T} \boldsymbol{K}^{-1}$ where $\boldsymbol{k}$ and $\boldsymbol{K}$ are commutative matrices and obey the matrix relation ${\boldsymbol{k}}^2=\boldsymbol{K}+3e_1\boldsymbol{I}+g{\boldsymbol{K}}^{-1}$. The obtained elliptic equations, both discrete and continuous, are formulated by the scalar function $S^{(i,j)}$ which is defined using $(\boldsymbol{k},\boldsymbol{K}, \boldsymbol{M}, \boldsymbol{r},\boldsymbol{c})$ and constitute an infinite size symmetric matrix. Lax pairs for both the discrete and continuous system are derived. The explicit solution $\boldsymbol{M}$ of the Sylvester equation and generalized solutions of the obtained elliptic equations are presented according to the canonical forms of matrix $\boldsymbol{k}$.

Abstract:
In this paper, we present a novel fast method to solve Poisson's equation in an arbitrary two dimensional region with Neumann boundary condition, which are frequently encountered in solving electrostatic boundary problems. The basic idea is to solve the original Poisson's equation by a two-step procedure. In the first stage, we expand the electric field of interest by a set of tree basis functions and solve it with a fast tree solver in O() operations. The field such obtained, however, fails to expand the exact field because the tree basis is not curl-free. Despite of this, we can retrieve the correct electric field by purging the divergence-free field. Next, for the second stage, we find the potential distribution rapidly with a same fast solution of O() complexity. As a result, the proposed method dramatically reduces solution time compared with traditional FEM with iterative method. In addition, it is the first time that the loop-tree decomposition technique has been introduced to develop fast Poisson solvers. Numerical examples including electrostatic simulations are presented to demonstrate the efficiency of the proposed method.

Abstract:
In this paper we present a novel fast method to solve Poisson equation in an arbitrary two dimensional region with Neumann boundary condition. The basic idea is to solve the original Poisson problem by a two-step procedure: the first one finds the electric displacement field $\mathbf{D}$ and the second one involves the solution of potential $\phi$. The first step exploits loop-tree decomposition technique that has been applied widely in integral equations within the computational electromagnetics (CEM) community. We expand the electric displacement field in terms of a tree basis. Then, coefficients of the tree basis can be found by the fast tree solver in O(N) operations. Such obtained solution, however, fails to expand the exact field because the tree basis is not completely curl free. Despite of this, the accurate field could be retrieved by carrying out a procedure of divergence free field removal. Subsequently, potential distribution $\phi$ can be found rapidly at the second stage with another fast approach of O(N) complexity. As a result, the method dramatically reduces solution time comparing with traditional FEM with iterative method. Numerical examples including electrostatic simulations are presented to demonstrate the efficiency of the proposed method.