General Numerical Scheme Research Articles

Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations

In this paper, we establish Razumikhin-type theorems on $$\alpha $$ th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global $$\alpha $$ th moment asymptotically stability and $$\alpha $$ th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler–Maruyama method and the backward Euler–Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on $$\alpha $$ th moment polynomial stability.

A three‐scale compressible microsphere model for hyperelastic materials

SummaryThis paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571‐592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg, stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three‐scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro‐meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three‐scale microsphere model is investigated through representative boundary value problems.

A general method for simulating reactive dissolution in carbonate rocks with arbitrary geometry

The two-scale continuum model is widely used in simulating the reactive dissolution process and predicting the optimum injection rate for carbonate reservoir acidizing treatment. The numerical methods of this model are currently based on structured grids, which are not applicable for complicated geometries. In this study, a general numerical scheme for simulating a reactive flow problem on both structured and unstructured grids is presented based on the finite volume method (FVM). The convection and diffusion terms involved in the reactive flow model are discretized by using the upwind scheme and two-point flux approximation (TPFA), respectively. The location of the centroid node inside each control volume is moved by using an optimization algorithm to make the connections with the surrounding elements as orthogonal as possible, which systematically improves the accuracy of the TPFA scheme. Additionally, in order to avoid the computational complexity resulting from the discretization of the non-linear term, the mass balance equation is only discretized in the spatial domain to get a set of ordinary differential equations (ODEs). These ODEs are coupled with the reaction equations and then solved using the numerical algorithm on ODEs. The accuracy and efficiency of the proposed method are studied by comparing the results obtained from the proposed numerical method with previous experimental and numerical results. This comparison indicates that, compared with the previous methods, the proposed method predicts the wormhole structure more accurately. Finally, the presented method is used to check the effect of the domain geometry, and it is found that the geometry of the flow domain has no effect on the optimum injection velocity, but the radial domain requires a larger breakthrough volume than the linear domain when other parameters are fixed.

A General Scheme for the Discontinuous Galerkin Time-Domain Modeling and S-Parameter Extraction of Inhomogeneous Waveports

A general numerical scheme is proposed for the waveport modeling and scattering-parameter (S-parameter) extraction of inhomogeneous waveports with the discontinuous Galerkin time-domain method. In this scheme, the waveports are truncated with perfectly matched layers with a hybrid mesh automatically extruded from the mesh of the physical device to be simulated. The waveports are then excited by a total-field/scattered-field technique, with which the incident and scattered waves can be obtained for an accurate calculation of the S-parameters. A novel eigenmode solver is also developed to calculate the required modal profiles for the eigenmodes in both homogeneous and inhomogeneous waveports. Special attention is paid to the S-parameter extraction for evanescent modes, which has been a difficult task for time-domain simulations. Numerical examples are given to validate the proposed numerical scheme.

Effects of Protein-Induced Local Bending and Sequence Dependence on the Configurations of Supercoiled DNA Minicircles.

The statistical mechanics of a short circularized DNA molecule, a DNA minicircle, with a prescribed linking number depends heavily on the mechanical and geometrical properties of the DNA, which are known to be functions of the sequence. A description of a general numerical scheme used for the performed advanced simulation is presented with examples that reveal the effects of sequence dependence and local deformations, caused by protein binding, on the configurations in a canonical ensemble of a supercoiled minicircle. Using a realistic course grain model in which the sequence-dependent elasticity, the intramolecular electrostatic interactions, and the impenetrability of the DNA molecule are taken into account, the bifurcation of equilibria of supercoiled minicircle DNA with the induced bending as a parameter is shown. The unique Monte Carlo scheme for constrained DNA molecules was utilized in order to calculate site-to-site distance probabilities for each pair of sites in the molecule. The simulated examples show not only that the sequence alone can play a significant role in bringing two remote sites to a contact but also the possible existence of several competing regions of contact. The results suggest that the local deformation caused by protein binding can yield a global configurational change, dominated by slithering motion, which brings two (originally) remote sites to close proximity, and that the nature of such effect is related to the sequence architecture.

Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost

An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order HJB equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. A convergent semi-Lagrangian scheme is presented in detail.

Modeling of supercritical-pressure turbulent combustion of hydrocarbon fuels using a modified flamelet-progress-variable approach

Non-premixed turbulent combustion at a supercritical pressure is an important physicochemical phenomenon in many propulsion and power-generation systems. In this paper, a modified flamelet-progress-variable model, in which a simple term is proposed to approximately account for the extra turbulent mixing related to large density difference between two fluid streams, has been developed and incorporated into a single-phase general fluid numerical scheme in RANS framework for solving supercritical-pressure turbulent combustion of hydrocarbon fuels. The model was validated and then applied for studying the coaxial injection and turbulent combustion of LOx/methane and the swirling injection and turbulent combustion of LOx/kerosene at various supercritical pressures. Results indicate that for the coaxial injection and combustion of LOx/methane at a mixture ratio of 3, flame length decreases as chamber pressure increases, dictated by the penetration capability of the injected LOx stream. For the swirling injection and combustion of LOx/kerosene at supercritical pressures, flame moves closer to the injector as chamber pressure increases. This phenomenon at a higher pressure is similar to that caused by an increased inlet-flow swirling number. It suggests that increasing chamber pressure may lead to a shorter combustor for the turbulent combustion of LOx and hydrocarbon fuels.

On a general numerical scheme for the fractional plastic flow rule

This paper presents a general numerical scheme for the fractional plastic flow rule, dedicated to a wide class of materials manifesting the non-normality of plastic flow and induced plastic anisotropy. To determine the vector of the plastic flow, a special numerical procedure has been developed, which is applicable for any smooth and convex yield function. The obtained approximation is verified based on an analytical solution. The paper also presents a set of numerical results for the generalised Drucker–Prager model.

Non-monotone algorithm for minimization on arbitrary domains with applications to large-scale orthogonal Procrustes problem

This paper concerns a non-monotone algorithm for minimizing differentiable functions on closed sets. A general numerical scheme is proposed which combines a regularization/trust-region framework with a non-monotone strategy. Global convergence to stationary points is proved under usual assumptions. Numerical experiments for a particular version of the general algorithm are reported. In addition, a promising numerical scheme for medium/large-scale orthogonal Procrustes problem is also proposed and numerically illustrated.

Elastohydrodynamic Lubrication in Ball Thrust Bearing

This paper presents the theoretical characteristics of elastohydrodynamic lubrication (EHL) in point contact under steady operating conditions of ball bearing thrust bearing. The numerical simulations employed a finite different method, Newton Raphson method and multigrid method to solve the modified Reynolds equation with a Non-Newtonian fluid. The general numerical schemes are implemented to investigate the profile of pressure and film thickness, with varying applied loads viscosity of lubricants and speeds. The results show that the applied load has significant effect on the film thickness profile. The contact is increase area but film thickness decrease as the applied load increase. The minimum film thickness and friction coefficient both increase significantly as viscosity of lubricant is increased. The increasing of surface velocity, the film thickness increase but film pressure decrease.

Convergence rate of free boundary of numerical scheme for American option

Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function ofan explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.

Detection of magnetic barriers in a chaotic domain: first application of finite time Lyapunov exponent method to a magnetic confinement configuration

Magnetic field lines embedded in a plasma confinement system are often characterized by a chaotic motion. This weakens the confinement properties of any magnetic configuration. However, even in case of chaotic domains, magnetic barriers can emerge and limit the field line motion itself. In the context of the numerical simulation of a Reversed-Field Pinch configuration a new magnetic topology analysis, borrowed from previous fluid dynamic studies, is discussed. This methodology relies on the behavior of the Finite Time Lyapunov Exponent (FTLE) associated with the magnetic field. By referring to a previous work in which the magnetic field is given in terms of analytical function (Borgogno et al 2011 Phys. Plasmas 18 102307) the FTLE field shows the presence of ridges, special gradient lines normal to the direction of minimum curvature, forming magnetic barriers. These ridges can be recognized as Lagrangian Coherent Structures (LCSs) for the system, actually opposing the penetration of magnetic field lines across them. In this article a more general numerical scheme for the detection of the LCSs has been adopted that allows analysis of realistic cases in which the magnetic fields are numerically known on a discrete mesh. After a validation test performed on the analytical case, a first application to a numerical magnetohydrodynamics simulation of the RFP, characterized by a broad chaotic region, has been performed. A strong magnetic barrier has been observed that effectively limits the field lines motion inside the chaotic sea.

Numerical solutions of the reaction-diffusion equation

Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton–Jacobi–Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

Probing the extensive nature of entropy

We have devised a general numerical scheme applied to a system of independent, distinguishable, non-interacting particles, to demonstrate in a direct manner the extensive nature of statistical entropy. Working within the microcanonical ensemble, our methods enable one to directly monitor the approach to the thermodynamic limit (N → ∞) in a manner that has not been known before. We show that (sN − s∞) → N−α where sN is the entropy per particle for N particles and S∞ is the entropy per particle in the thermodynamic limit. We demonstrate universal behaviour by considering a number of different systems each defined by its unique single-particle spectrum. Various thermodynamic quantities as a function of N may be computed using our methods; in this paper, we focus on the entropy, the chemical potential and the temperature. Our results are applicable to systems of finite size, e.g. nano-particle systems. Furthermore, we demonstrate a new phenomenon, referred to as entropic interference, which manifests as a cancellation of terms in the thermodynamic limit and which results in the additive nature of entropy.

SOR waveform relaxation methods for stochastic differential equations

Waveform relaxation for solving large-scale systems of ordinary differential equations has extended to stochastic differential equations (SDEs). This paper deals with the acceleration of the Gauss–Sediel waveform relaxation method for SDEs by successive overrelaxation (SOR) techniques. A sufficient condition of the linear convergence is obtained for the continuous time SOR waveform relaxation method. The discrete time SOR waveform relaxation method used in an actual implementation was also studied. We first prove that the general numerical schemes are convergent and then the particular method based on Euler scheme linearly converge to the Euler approximate solution of SDEs. At last, the theory is applied to a model problem and checked against results obtained by numerical experiments.

Numerical Algorithms for J<sub>2</sub> Viscoplastic Models

An algorithmic scheme is illustrated for the numerical integration of evolutive problems in viscoplasticity. The analysis is restricted for brevity to J2 models in viscoplasticity by presenting a unified treatment of different elasto/viscoplastic constitutive models. An appropriate consistent tangent operator is applied which is suitable to be adopted for general numerical schemes. The treatment can be applied to different constitutive models by simply specializing the flow function. Computational results and examples are also illustrated in order to show the effectiveness of the presented algorithmic procedure.

Pore-scale simulation of coupled multiple physicochemical thermal processes in micro reactor for hydrogen production using lattice Boltzmann method

A general numerical scheme based on the lattice Boltzmann method (LBM) is established to investigate coupled multiple physicochemical thermal processes at the pore-scale, in which several sets of distribution functions are introduced to simulate fluid flow, mass transport, heat transfer and chemical reaction. Interactions among these processes are also considered. The scheme is then employed to study the reactive transport in a posted micro reactor. Specially, ammonia (NH3) decomposition, which can generate hydrogen (H2) for fuel of proton exchange membrane fuel cells (PEMFCs), is considered where the endothermic decomposition reaction takes place at the surface of posts covered with catalysts. Simulation results show that pore-scale phenomena are well captured and the coupled processes are clearly predicted. Effects of several operating and geometrical conditions including NH3 flow rate, operating temperature, post size, post insert position, post orientation, post arrangement and post orientation on the coupled physicochemical thermal processes are assessed in terms of NH3 conversion, temperature uniformity, H2 flow rate and subsequent current density generated in PEMFC.

Numerical resolution of a mono-disperse model of bubble growth in magmas

Growth of gas bubbles in magmas may be modeled by a system of differential equations that account for the evolution of bubble radius and internal pressure and that are coupled with an advection–diffusion equation defining the gas flux going from magma to bubble. This system of equations is characterized by two relaxation parameters linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme preserving, by construction, the total mass of water of the system. We also study the asymptotic behavior of the system of equations by letting the relaxation parameters vary from 0 to ∞, and show the numerical convergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments.

Fractional optimal control of distributed systems in spherical and cylindrical coordinates

This paper presents a general formulation and numerical scheme for the fractional optimal control problem (FOCP) of distributed systems in spherical and cylindrical coordinates. The fractional derivatives are expressed in the Caputo-Sense. The performance index of FOCP is considered as a function of both the state and the control variables and the dynamic constraints are expressed by a partial fractional differential equation. A method of separation of variables is employed to separate the time and space terms, and the eigenfunction approach is used to eliminate the terms containing space parameter and define the formulation in terms of countable number of infinite state and control variables. The fractional optimal control equations are reduced to the Volterra-type integral equations. For the problems considered, only a few eigenfunctions in each direction are sufficient for the calculations to converge. The time domain is discretized into several subintervals and the result is more stable for a larger number of time segments. Various orders of fractional derivatives are analyzed and the results converge toward those of integer optimal control problems as the order approaches the integer value of 1.