Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

In order to solve the partial differential equations that arise in the Hartree-Fock theory for diatomic molecules and in molecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variable model problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9-point difference formula produces the same level of accuracy with fewer points. The domain decomposition method has the strength that it can be applied to problems with a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments thatare calculated independently. Since the domain decomposition approach makes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.     

Solutions of the 1D Coupled Nonlinear Schrödinger Equations by the CIP-BS Method

In this paper, we present solutions for the one-dimensional coupled nonlinear Schrödinger (CNLS) equations by the Constrained Interpolation Profile-Basis Set (CIP-BS) method. This method uses a simple polynomial basis set, by which physical quantities are approximated with their values and derivatives associated with grid points. Nonlinear operations on functions are carried out in the framework of differential algebra. Then, by introducing scalar products and requiring the residue to be orthogonal to the basis, the linear and nonlinear partial differential equations are reduced to ordinary differential equations for values and spatial derivatives. The method gives stable, less diffusive, and accurate results for the CNLS equations.     

Successive approximation and optimal controls on fractional neutral stochastic differential equations with Poisson jumps

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations with Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations with Poisson jumps is also presented. An example is given to illustrate the theory. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of avoidance regions for driver assistance systems by using a Hamilton-Jacobi approach

We consider the problem of computing safety regions, modeled as nonconvex backward reachable sets, for a nonlinear car collision avoidance model with time-dependent obstacles. The Hamilton-Jacobi-Bellman framework is used. A new formulation of level set functions for obstacle avoidance is given, and sufficient conditions for granting the obstacle avoidance on the whole time interval are obtained even though the conditions are checked only at discrete times. Different scenarios, including various road configurations, different geometry of vehicle and obstacles, as well as fixed or moving obstacles, are then studied and computed. Computations involve solving nonlinear partial differential equations of up to five space dimensions plus time with nonsmooth obstacle representations, and an efficient solver is used to this end. A comparison with a direct optimal control approach is also done for one of the examples.     

The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions

In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations. We first approximate the source function by Chebyshev polynomials. We then focus on how to find a polynomial particular solution when the source function is a polynomial. Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined, the coefficients of the particular solution satisfy a triangular system of linear algebraic equations. Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs. The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs. Numerical results show that our approach is efficient and accurate.     

A Modified Nonlocal Continuum Electrostatic Model for Protein in Water and Its Analytical Solutions for Ionic Born Models

A nonlocal continuum electrostatic model, defined as integro-differential equations, can significantly improve the classic Poisson dielectric model, but is too costly to be applied to large protein simulations. To sharply reduce the model's complexity, a modified nonlocal continuum electrostatic model is presented in this paper for a protein immersed in water solvent, and then transformed equivalently as a system of partial differential equations. By using this new differential equation system, analytical solutions are derived for three different nonlocal ionic Born models, where a monoatomic ion is treated as a dielectric continuum ball with point charge either in the center or uniformly distributed on the surface of the ball. These solutions are analytically verified to satisfy the original integro-differential equations, thereby, validating the new differential equation system.     

Godunov-Type Numerical Methods for a Model of Granular Flow on Open Tables with Walls

We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.     

Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equation, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to the estimation of a physical length scale in the case of a uniform static grid.     

Trajectory optimization involving sloshing media

This paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two‐dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non‐linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.     

A Riccati-equation-based algorithm for continuous-time optimal control problems

In this paper we consider continuous-time unconstrained optimal control problems. We propose a computational method which is essentially based on the closed-loop solutions of the linear quadratic optimal control problems. In the proposed algorithm, Riccati differential equations play an important role. We prove that accumulation points generated by the present algorithm, if they exist, satisfy the weak necessary conditions for optimality, under some assumptions including Kalman's sufficient conditions for the bounded Riccati solutions. In addition, we also propose the simple but effective technique to guarantee the boundedness of the solutions of Riccati equations. Lastly, we illustrate the usefulness of the present algorithm through simulation experiences. Copyright © 1998 John Wiley & Sons, Ltd.     

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.     

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.     

Mechanics-Based Solution Verification for Porous Media Models

This paper presents a new approach to verify the accuracy of computational simulations. We develop mathematical theorems which can serve as robust a posteriori error estimation techniques to identify numerical pollution, check the performance of adaptive meshes, and verify numerical solutions. We demonstrate performance of this methodology on problems from flow thorough porous media. However, one can extend it to other models. We construct mathematical properties such that the solutions to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties include the total minimum mechanical power, minimum dissipation theorem, reciprocal relation, and maximum principle for the vorticity. All the developed theorems have firm mechanical bases and are independent of numerical methods. So, these can be utilized for solution verification of finite element, finite volume, finite difference, lattice Boltzmann methods and so forth. In particular, we show that, for a given set of boundary conditions, Darcy velocity has the minimum total mechanical power of all the kinematically admissible vector fields. We also show that a similar result holds for Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and Darcy-Brinkman velocities have the minimum total dissipation among their respective kinematically admissible vector fields. Using numerical examples, we show that the minimum dissipation and total mechanical power theorems can be utilized to identify pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman equations are shown to satisfy a reciprocal relation, which has the potential to identify errors in the numerical implementation of boundary conditions. It is also shown that the vorticity under both steady and transient Darcy-Brinkman equations satisfy maximum principles if the body force is conservative and the permeability is homogeneous and isotropic. A discussion on the nature of vorticity under steady and transient Darcy equations is also presented. Using several numerical examples, we will demonstrate the predictive capabilities of the proposed a posteriori techniques in assessing the accuracy of numerical solutions for a general class of problems, which could involve complex domains and general computational grids.     

Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System

Many partial differential equations can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.     

Lattice Boltzmann Schemes with Relative Velocities

In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d'Humières. They extend also the Geier's cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.     

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

In this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.     

GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.     

Optimal control of temperature and fuel consumption for soaking pits in a steel plant—a distributed parameter approach

An optimal control theory for a class of non-linear distributed parameter control systems in the general setting involving more than one spatial co-ordinate is developed for use on control of a typical soaking pit in a steel industry. The system is described by a set of non-linear partial differential equations in multidimensional spatial co-ordinates and non-linear boundary conditions with time-dependent boundary controls as well as a spatially independent parameter vector which is governed by its own set of dynamic equations. The set of necessary conditions for optimality obtained from the theoretical development is directly applied to the optimal heating control of the soaking pit with rectangular ingots. The numerical solution by iterations demonstrates the success of the technique and the algorithm leading to the optimal policy for heating control of a typical soaking pit with minimum fuel consumption.     

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.     

On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero-sum linear quadratic stochastic differential game

We study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero-sum two-player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game-theoretic Riccati differential equations arising in the solution of the H_∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero-sum LQ stochastic differential game on an infinite-time horizon and give as a byproduct the solution of such a control problem.