Topic

# Poisson's equation

About: Poisson's equation is a research topic. Over the lifetime, 7280 publications have been published within this topic receiving 132085 citations. The topic is also known as: Poisson equation & Poisson differential equation.

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TL;DR: In this paper, a moving-particle semi-implicit (MPS) method for simulating fragmentation of incompressible fluids is presented, where the motion of each particle is calculated through interactions with neighboring particles covered with the kernel function.

Abstract: A moving-particle semi-implicit (MPS) method for simulating fragmentation of incompressible fluids is presented. The motion of each particle is calculated through interactions with neighboring particles covered with the kernel function. Deterministic particle interaction models representing gradient, Laplacian, and free surfaces are proposed. Fluid density is implicitly required to be constant as the incompressibility condition, while the other terms are explicitly calculated. The Poisson equation of pressure is solved by the incomplete Cholesky conjugate gradient method. Collapse of a water column is calculated using MPS. The effect of parameters in the models is investigated in test calculations. Good agreement with an experiment is obtained even if fragmentation and coalescence of the fluid take place.

1,653 citations

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TL;DR: In this paper, an analytical solution of the linearized Poisson-Boltzmann (PB) equation valid in a spherical region is obtained, which can be used for evaluating the electrostatic potential and its derivative at the origin of the sphere.

Abstract: Molecular dynamics simulations of ionic systems require the inclusion of long‐range electrostatic forces. We propose an expression for the long‐range electrostatic forces based on an analytical solution of the Poisson–Boltzmann equation outside a spherical cutoff, which can easily be implemented in molecular simulation programs. An analytical solution of the linearized Poisson–Boltzmann (PB) equation valid in a spherical region is obtained. From this general solution special expressions are derived for evaluating the electrostatic potential and its derivative at the origin of the sphere. These expressions have been implemented for molecular dynamics (MD) simulations, such that the surface of the cutoff sphere around a charged particle is identified with the spherical boundary of the Poisson–Boltzmann problem. The analytical solution of the Poisson–Boltzmann equation is valid for the cutoff sphere and can be used for calculating the reaction field forces on the central charge, assuming a uniform continuum of given ionic strength beyond the cutoff. MD simulations are performed for a periodic system consisting of 2127 SPC water molecules with 40 NaCl ions (1 molar). We compare the structural and dynamical results obtained from MD simulations in which the long range electrostatic interactions are treated differently; using a cutoff radius, using a cutoff radius and a Poisson–Boltzmann generalized reaction field force, and using the Ewald summation. Application of the Poisson–Boltzmann generalized reaction field gives a dramatic improvement of the structure of the solution compared to a simple cutoff treatment, at no extra computational cost.

1,311 citations

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TL;DR: In this paper, a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions is presented, and it is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached.

Abstract: We construct a converging adaptive algorithm for linear elements applied to Poisson’s equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extension to higher-order elements in two space dimension and numerical results are included.

1,255 citations

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TL;DR: Using generic interpolation machinery based on solving Poisson equations, a variety of novel tools are introduced for seamless editing of image regions as discussed by the authors, and the first set of tools permits the seamless...

Abstract: Using generic interpolation machinery based on solving Poisson equations, a variety of novel tools are introduced for seamless editing of image regions. The first set of tools permits the seamless ...

1,183 citations

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TL;DR: The relationship between the actual charge and the renormalized charge by solving the Boltzmann-Poisson equation numerically in a spherical Wigner-Seitz cell was analyzed in this article.

Abstract: The interactions between charged colloidal particles with sufficient strength to cause crystallization are shown to be describable in terms of the usual Debye–Huckel approximation, but with a renormalized charge. The effective charge in general is smaller than the actual charge. We calculate the relationship between the actual charge and the renormalized charge by solving the Boltzmann–Poisson equation numerically in a spherical Wigner–Seitz cell. We then relate the numerical solutions and the effective charge to the osmotic pressure and the bulk modulus of the crystal. Our calculations also reveal that the renormalization of the added electrolyte concentration is negligible, so that the effective charge computations are useful even in the presence of salts.

827 citations