Poisson-type Equation Research Articles

A positivity-preserving and free energy dissipative hybrid scheme for the Poisson-Nernst-Planck equations on polygonal and polyhedral meshes

This paper develops a positivity-preserving and free energy dissipative hybrid scheme for the multi-dimensional Poisson-Nernst-Planck equations with multiple ions on unstructured polygonal and polyhedral meshes. It is built on the linear virtual element method for the Poisson-type equation for the electrostatic potential and a cell-centered finite volume scheme for the Nernst-Planck equations for the ion concentrations, and preserves the positivity of the ion concentrations, the mass conservation, the free energy dissipation and steady-state properties. The existence of the numerical solutions is also discussed. Numerical results demonstrate the effectiveness of the proposed two- and three-dimensional schemes.

Linear and nonlinear non-Fredholm operators and their applications

<abstract><p>In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.</p></abstract>

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method.
 Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis

We construct new first- and second-order pressure correctionschemes using the scalar auxiliary variable approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates in the two dimensional case for the velocity and pressure approximation of the first-order scheme without any condition on the time step.

Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup

This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.

Simple and efficient volume merging method for triply periodic minimal structures

Triply periodic minimal surfaces (TPMSs), which are periodic in all three directions and are surfaces of zero mean curvature, have been proven experimentally to be highly suitable for tissue scaffolds. However, simply gluing different TPMS units with different porosities and pore sizes could induce discontinuous structures and destroy the physical properties. In this study, we propose a simple and efficient volume merging method for triply periodic minimal structures. The proposed method can be divided into two steps. The first step is a novel merging algorithm for unit triply periodic minimal structures in the implicit function framework. The composite scaffold can be designed by merging different unit structures to satisfy the properties of internal connectivity. The second step is to optimize the designed composite scaffolds to satisfy the properties of TPMSs. A modified Allen–Cahn-type equation with a correction term is proposed. The mean curvature on the surface is constant at all points in the equilibrium state. Typically, the obtained structure is smooth owing to the motion by mean curvature flow. Therefore, the quality of the structure is significantly improved. Based on the operator splitting method, the proposed algorithm consists of two analytical evaluations for the ordinary differential equations and one numerical solution for the implicit Poisson-type equation. The proposed numerical scheme can be applied in a straightforward manner to a GPU-accelerated discrete cosine transform (DCT) implementation, which can be executed multiple times faster than CPU-only alternatives. Computational experiments are presented to demonstrate the efficiency and robustness of the proposed method.

On the evaluation of Poisson equation with dual interpolation boundary face method

This paper presents a new implementation of the dual reciprocity method (DRM) in connection with the dual interpolation boundary face method (DiBFM) for the Poisson equation. In DiBFM, the nodes of an element are categorized into two groups: (i) source nodes (ii) virtual nodes. First layer interpolation is used to interpolate the physical variables, while boundary integrals are evaluated on the source nodes only. Moreover, moving least squares (MLS) interpolation is used and provides additional constraints equations to establish the relationship between source and virtual nodes. Additionally, augmented thin plate spline (ATPS) is used to better interpolate the non-homogeneous term. Finally, it is claimed that the proposed method is much superior to the DRM for Poisson type equation with different geometries, especially for complex geometry. Numerical examples are evaluated and compared with the DRM to ensure the superiority of the proposed method.

An unstructured finite-volume semi-coupled projection model for bed load sediment transport in shallow-water flows

The aim of this work is to develop an original semi-coupled approach for solving the Saint-Venant and Exner equation system. The shallow-water equations are solved first to calculate the full evolution of flow fields, and then the Exner equation is solved to model the morphodynamic response. The numerical method is based on a projection method, which consists in combining the momentum and continuity equations in order to establish a Poisson-type equation for water surface levels. The present formulation is identified as a semi-coupled approach because the projection method incorporates the change in the bed evolution in its approximation. A second-order numerical scheme has been proposed using an unstructured finite-volume technique and an implicit time integration method. Several benchmarks are used to demonstrate the capabilities, accuracy and performance of the model.

The Dirichlet problem for the Poisson type equations in the plane

The Dirichlet problem for the Poisson type equations in the plane

A gPAV-based unconditionally energy-stable scheme for incompressible flows with outflow/open boundaries

We present an unconditionally energy-stable scheme for approximating the incompressible Navier–Stokes equations on domains with outflow/open boundaries. The scheme combines the generalized Positive Auxiliary Variable (gPAV) approach and a rotational velocity-correction type strategy, and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions. The discrete energy stability of the proposed scheme has been proven, irrespective of the time step sizes. Within each time step the scheme entails the computation of two velocity fields and two pressure fields, by solving an individual de-coupled Helmholtz (including Poisson) type equation with a constant pre-computable coefficient matrix for each of these field variables. The auxiliary variable, being a scalar number, is given by a well-defined explicit formula within a time step, which ensures the positivity of its computed values. Extensive numerical experiments with several flows involving outflow/open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes.

Fast Fourier Solvers for the Tensor Product High-Order FEM for a Poisson Type Equation

Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the N-dimensional Poisson-type equation $$ - \Delta u + \alpha u = f$$ ($$N \geqslant 2$$) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrodinger ones.

An efficient and accurate MPI-based parallel simulator for streamer discharges in three dimensions

In this paper, we propose an efficient and accurate message-passing interface (MPI)-based parallel simulator for streamer discharges in three dimensions using the fluid model. First, we propose a new second-order semi-implicit scheme for the temporal discretization of the model that relaxes the dielectric relaxation time restriction. Moreover, it requires solving the Poisson-type equation only once at each time step, while the classical second-order explicit schemes typically need to do twice. Second, we introduce a geometric multigrid preconditioned FGMRES solver that dramatically improves the efficiency of solving the Poisson-type equation with either constant or variable coefficients. We show numerically that no more than 4 iterations are required for the Poisson solver to converge to a relative residual of 10−8 during streamer simulations; the FGMRES solver is much faster than R&B SOR and other Krylov subspace solvers. Last but not least, all the methods are implemented using MPI. The parallel efficiency of the code and the fast algorithmic performances are demonstrated by a series of numerical experiments using up to 2560 cores on the Tianhe2-JK clusters. For applications, we study a double-headed streamer discharge as well as the interaction between two streamers, using up to 10.7 billion mesh cells.

Stochastic Modelling for Dynamics of Interacting Populations

Abstract The paper deals with a mathematical model for two interacting populations. Under the assumption of fast stochastic contacting of populations, we derive stochastic Poisson-type differential equations with a small parameter and propose an approximative algorithm for quantitative analysis of population dynamics that consists of two steps. First, we derive an ordinary differential equation for mean value of each population growth and analyse the average asymptotic population behaviour. Then, applying diffusion approximation procedure, we derive a stochastic Ito differential equation for small random deviations on the average motion in a form of a linear non-homogeneous Ito stochastic differential equation and analyse the probabilistic characteristics of the Gaussian process given by this equation.

Existence Results of Two Mixed Boundary Value Elliptic PDEs in $$\mathbb {R}^N$$

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $$\Omega \subset \mathbb {R}^N$$ and in $$\mathbb {R}^N{\setminus }\Omega $$ for $$N\ge 3$$. The boundary $$\partial \Omega $$ of $$\Omega $$ is the decomposition of $$\Gamma _1,\Gamma _2\subset \partial \Omega $$ such that $$\partial \Omega =\Gamma =\overline{\Gamma }_1\cup \Gamma _2=\Gamma _1\cup \overline{\Gamma }_2$$ and $$\Gamma _1\cap \Gamma _2=\emptyset $$. We have shown that if the Neumann data $$f_2\in H^{-\frac{1}{2}}(\Gamma _2)$$ and the Dirichlet data $$f_1\in H^{\frac{1}{2}}(\Gamma _1)$$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to $$f_1\in H^{\frac{1}{2}}(\Gamma _1)$$, $$f_2\in H^{-\frac{1}{2}}(\Gamma _2)$$, respectively.

Numerical Study of Laminar Natural Convection Heat Transfer in Inclined Trapezoidal Enclosure

In this paper, fluid flow pattern and heat transfer behavior are numerically studied in a trapezoidal enclosure inclined at different tilt angles. The enclosure has isothermally heated cylindrical inner wall and cooled cylindrical outer wall, whereas sidewalls have linearly varying temperature profile. The numerical analysis is performed at different Rayleigh numbers (103 ≤ Ra ≤106) and Prandtl numbers (0.71 ≤ Pr ≤ 40) assuming steady, incompressible, laminar, Newtonian, and two-dimensional flow. The angle between two nonparallel sidewalls is varied from 10 deg to 90 deg, while the tilt angle is varied from 5 deg to 175 deg. Governing equations along with the Poisson type equation for heatfunctions are solved to obtain fluid flow and direction of heat flow in the trapezoidal enclosure. At low Raleigh number, the isotherms are smooth, indicating the dominance of conduction heat transfer dominates. The streamlines, isotherms, and heatlines in high Prandtl number (Pr = 6.99 ad 40) fluids show a similar pattern as that in the air (Pr = 0.71) at Ra = 106; however, the strength of heatfunctions increases. With the increase in the aspect ratio of the enclosure, thermally stratified region is found to increase in size. Stronger convection is present in the enclosure with a higher angle between two nonparallel sidewalls.

An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces

We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting proposed in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955–980] in the context of isentropic Euler and Navier-Stokes equations. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver (we use the second-order central-upwind scheme) to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme.We conduct a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.

Solving 2D Poisson-type equations using meshless SPH method

Abstract In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Compared with other kernel functions, the cubic B-spline kernel function shows good capacity to reproduce a complicated function value and its first and second derivatives, when the smoothing length is chosen as 1.1 times the particles’ distance. Several types of Poisson equations are solved by SSPH method, and the numerical results exhibit a very good accuracy when comparing with the analytical solutions.

Multiple positive solutions of a class of Schrödinger–Poisson equation involving indefinite nonlinearity in [formula omitted

In this paper, we study the existence of multiple positive solutions of Schrödinger–Poisson type equations with indefinite nonlinearity. Our main tool is the mountain pass theorem.

Solving Poisson-type equations with Robin boundary conditions on piecewise smooth interfaces

We present two finite volume schemes to solve a class of Poisson-type equations subject to Robin boundary conditions in irregular domains with piecewise smooth boundaries. The first scheme results in a symmetric linear system and produces second-order accurate numerical solutions with first-order accurate gradients in the L∞-norm (for solutions with two bounded derivatives). The second scheme is nonsymmetric but produces second-order accurate numerical solutions as well as second-order accurate gradients in the L∞-norm (for solutions with three bounded derivatives). Numerical examples are given in two and three spatial dimensions.

Fast Fourier transform-based micromechanics of interfacial line defects in crystalline materials

Spectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics community. The present contribution addresses the critical question of determining local mechanical fields using the FFT method in the presence of interfacial defects. Precisely, the present work introduces a numerical approach based on intrinsic discrete Fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of disclinations, i.e., rotational discontinuities, and inhomogeneities. A centered finite difference scheme for differential rules are first used for numerically solving the Poisson-type equations in the Fourier space to get the incompatible elastic fields due to disclinations and dislocations. Second, centered finite differences on a rotated grid are chosen for the computation of the modified Fourier-Green’s operator in the Lippmann–Schwinger–Dyson type equation for heterogeneous media. Elastic fields of disclination dipole distributions interacting with inhomogeneities of varying stiffnesses, grain boundaries seen as DSUM (Disclination Structural Unit Model), grain boundary disconnection defects and phase boundary “terraces” in anisotropic bi-materials are numerically computed as applications of the method.