Iterative Finite Difference Scheme Research Articles

A fictitious-play finite-difference method for linearly solvable mean field games

An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

In this article, we developed a new higher-order implicit finite difference iterative scheme (FDIS) for the solution of the two dimension (2-D) time fractional Cable equation (FCE). In the new proposed FDIS, the time fractional and space derivatives are discretized using the Caputo fractional derivative and fourth-order implicit scheme, respectively. Moreover, the proposed scheme theoretical analysis (convergence and stability) is also discussed using the Fourier analysis method. Finally, some numerical test problems are presented to show the effectiveness of the proposed method.

Solution of Unsteady Flow Equations for Nanoscale Heat and Mass Transfer in Diverse Applications

Abstract A new innovative stable time-dependent compressible flow solution over the order of nanoseconds is provided here for wide-ranging critically important challenging applications. Specifically, a solution of the highly complex unsteady high speed oscillating compressible flow field inside a cylindrical tube, closed at one end with a piston oscillating at very high resonant frequency at the other end, has been obtained numerically, assuming one-dimensional, viscous, and heat-conducting flow, by solving the appropriate fluid dynamic and energy equations. An iterative implicit finite difference scheme is employed to obtain the solution. The scheme permits arbitrary boundary conditions at the piston and the end wall and allows assumptions for transport properties. In successfully predicting the time-dependent results/data, an innovative simple but stable solution of unsteady fluid dynamic and energy equations is provided here for wide ranging research, design, development, analysis, and industrial applications in solving a variety of complex fluid flow heat transfer problems. The method is directly applicable to pulsed or pulsating flow and wave motion thermal energy transport, fluid-structure interaction heat transfer enhancement, nanoscale heat and mass transfer, diverse range of advanced fluidics, biofluidics/bio-engineering, and fluidic pyrotechnic initiation devices. It can further be easily extended to cover muzzle blasts and high energy nuclear explosion blast wave propagations in one-dimensional and/or radial spherical coordinates with or without including energy generation/addition terms. No other solution exists for such applications.

A reduced-order extrapolated finite difference iterative scheme for uniform transmission line equation

In order to find the numerical solutions of the two-dimensional (2D) uniform transmission line equation, we first establish a fully second-order finite difference (FD) scheme in matrix-form and analyze the stability and convergence of the FD solutions by matrix analysis. Then we employ a proper orthogonal decomposition (POD) method to establish a reduced-order extrapolated FD (ROEFD) scheme containing very few unknowns for the 2D uniform transmission line equation and also make use of the matrix analysis to analyze the stability and convergence for the ROEFD solutions. Finally, we employ some numerical experiments to validate the effectiveness of the ROEFD scheme.

Electrohydrodynamics and thermal radiation effects on natural convection flow in an enclosed domain

The flow and heat transfer of electrohydrodynamics (EHD) natural convection in the annulus between a rectangular and a circular cylinder is studied numerically. The Rosseland diffusion approximation is used to account for the influence of heat radiation. The inner circular cylinder is considered to be heated and charged with a constant direct current (DC) voltage producing the electric field in the closed cavity. The fundamental Navier-Stokes equations and energy equation are coupled with Maxwell's and Gauss's law to establish the interaction between the velocity, temperature, and electric field for the dielectric fluid. While formulation, we presume that the charged fluid is ideal so that the effect of magnetic induction is negligible. A couple of transformations are used to convert the governing equations into the most appropriate form suitable for implementing the iterative finite difference scheme. According to the findings, the electric field and thermal radiations have a considerable impact on the flow field within the cavity. For instance, an increment in the charge diffusivity number, and the radiation parameter, the intensity of the streamlines, enhances significantly. The isotherms and isolines of electric charge density provide credible information about the heat flow and influence of charge injection. Moreover, the locally distributed Nusselt number at the inner and outer cylinders confirms that thermal radiation is the dominant mode of heat transfer.

Thermofluidic characteristics of electrokinetic flow in a rotating microchannel in presence of ion slip and Hall currents

A theoretical investigation has been carried out for electrokinetic flow and heat transfer in a rotating microchannel. The fluid is driven by a static electric field applied externally along the longitudinal direction of the microchannel. The entire flow field is exposed to a uniform magnetic field applied in the transverse direction. The heat generated due to electromagnetism and viscous dissipation effects are taken into account while studying the associated heat transfer problem. The governing equations for the electric potential, velocities and flow rates are solved analytically using the Debye-Hückel approximation, whereas a numerical method based on the finite difference iterative scheme is used to solve the associated non-dimensional heat equation. It is observed that the hydrodynamic boundary layers are reduced and a precipitous behaviour is found in the EDL region of the microchannel when it rotates at high speed. It has been found that the Hall parameter reduces the fluidic temperature at small magnitude of the Hartmann number (Ha = 1), while an opposite trend is depicted for larger Ha. Asymptotic behaviour of the Nusselt number is analyzed for a wide range of Brinkman number and Joule heating parameter.

Numerical solution of a third-order Falkner-Skan-like equation arising in boundary layer theory.

We consider the numerical solution of a third-order Falkner-Skan-like boundary value problem arising in boundary layer theory. The problem is defined on a semi-infinite interval [0,∞) with a condition given at ∞. We first transform the problem into a second-order boundary value problem defined on a finite interval [γ,1]. To solve the resulting boundary value problem, we developed an iterative finite-difference scheme based on Newton's quasilinearization. At every step, the linearized differential equation is approximated using the finite-difference method. Numerical results will be presented to demonstrate the efficiency of the method and will be compared with other results presented in the literature.

Solution of Unsteady Fluid Dynamic and Energy Equations for High-Speed Oscillating Compressible Flows and Blast Wave Propagations

Abstract A solution of the highly complex unsteady high-speed oscillating compressible flow field inside a cylindrical tube has been obtained numerically, assuming one-dimensional, viscous, and heat conducting flow, by solving the appropriate fluid dynamic and energy equations. The tube is approximated by a right circular cylinder closed at one end with a piston oscillating at very high resonant frequency at the other end. An iterative implicit finite difference scheme is employed to obtain the solution. The scheme permits arbitrary boundary conditions at the piston and the end wall and allows assumptions for transport properties. The solution would also be valid for tapered tubes if the variations in the cross-sectional area are small. In successfully predicting the time-dependent results, an innovative simple but stable solution of unsteady fluid dynamic and energy equations is provided here for wide-ranging research, design, development, analysis, and industrial applications in solving a variety of complex fluid flow heat transfer problems. The method is directly applicable to pulsed or pulsating flow and wave motion thermal energy transport, fluid–structure interaction heat transfer enhancement, and fluidic pyrotechnic initiation devices. It can further be easily extended to cover muzzle blasts and nuclear explosion blast wave propagations in one-dimensional and/or radial spherical coordinates with or without including energy generation/addition terms.

High-order compact scheme for the two-dimensional fractional Rayleigh\u2013Stokes problem for a heated generalized second-grade fluid

In this article, an unconditionally stable compact high-order iterative finite difference scheme is developed on solving the two-dimensional fractional Rayleigh–Stokes equation. A relationship between the Riemann–Liouville (R–L) and Grunwald–Letnikov (G–L) fractional derivatives is used for the time-fractional derivative, and a fourth-order compact Crank–Nicolson approximation is applied for the space derivative to produce a high-order compact scheme. The stability and convergence for the proposed method will be proven; the proposed method will be shown to have the order of convergence O(tau + h^{4}). Finally, numerical examples are provided to show the high accuracy solutions of the proposed scheme.

A reduced-order energy-stability-preserving finite difference iterative scheme based on POD for the Allen-Cahn equation

We focus on proposing and analyzing a reduced-order finite difference (ROFD) iterative scheme based on a proper orthogonal decomposition (POD) technique for Allen-Cahn equations with a small perturbation parameter and strong nonlinearity. We prove the discrete maximum-principle (DMP) preserving and discrete energy-stability (DES) preserving of the ROFD iterative solutions under a reasonable restriction on the time step size. We also analyze the convergence of the ROFD iterative solutions for Allen-Cahn equations. Numerical tests are presented to verify our proposed ROFD method, such as error estimates, convergence rates, DMP-preserving and DES-preserving. Moreover, it is shown that the computing time of ROFD iterative schemes with very few unknowns is much less than that of usual finite difference (FD) iterative schemes.

Soil–Structure Interaction of Plates on Extensible Geosynthetic-Stone Column Reinforced Earth Beds

A mechanical model has been presented for analysis of plates resting on extensible geosynthetic reinforced granular fill lying on soft soil improved by stone columns. Foundation has been idealised as a plate and geosynthetic layer as rough elastic membrane. Granular fill, soft soil and stone columns has been idealised respectively as Pasternak shear layer, Kelvin–Voigt elements and Winkler springs and their nonlinear behaviour has been considered. Deformation compatibility condition at interface of geosynthetic and granular fill has been considered. An iterative finite difference scheme with proper boundary conditions has been used to solve governing differential equations and the system of simultaneous equations so obtained, have been solved using LU decomposition method. Response of plate in terms of deflection and bending moment and tension mobilized in geosynthetic has been presented in non-dimensional form. Influence of type of geosynthetic (extensible and inextensible) in reducing the settlements in conjunction with stone columns has been presented and the significance of each parameter influencing the plate response has been studied by means of detailed parametric study.

An iterative finite difference method for approximating the two-branched solution of Bratu's problem

In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ, where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme.

A one-dimensional nonlinear problem of thermoelasticity in extended thermodynamics

Abstract We solve a nonlinear, one-dimensional initial boundary-value problem of thermoelasticity in generalized thermodynamics. A Cattaneo-type evolution equation for the heat flux is used, which differs from the one used extensively in the literature. The hyperbolic nature of the associated linear system is clarified through a study of the characteristic curves. Progressive wave solutions with two finite speeds are noted. A numerical treatment is presented for the nonlinear system using a three-step, quasi-linearization, iterative finite-difference scheme for which the linear system of equations is the initial step in the iteration. The obtained results are discussed in detail. They clearly show the hyperbolic nature of the system, and may be of interest in investigating thermoelastic materials, not only at low temperatures, but also during high temperature processes involving rapid changes in temperature as in laser treatment of surfaces.

Nonlinear theory for thermoelastic solids with mass diffusion

Abstract This paper is concerned with a nonlinear second order thermoelastic diffusion theory for anisotropic centrosymmetric and homogeneous materials. The governing equations are presented in material form, which is especially adequate in dealing with moving boundaries. The case of isotropic materials is derived. We carry out some numerical simulation to validate and to show the capability of the developed model. The one-dimensional problem is treated numerically by a three-step, iterative finite-difference scheme. Comparisons are made with the results predicted by the thermoelastic problem without diffusion.

A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation

In this study, we devote ourselves to the reduced-order extrapolated finite difference iterative (ROEFDI) modeling and analysis for the two-dimensional (2D) Sobolev equation. To this end, we first establish the reduced-order extrapolated finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the 2D Sobolev equation via the proper orthogonal decomposition (POD) technique. And then, we analyze the stability and convergence of the ROEFDI solutions. Finally, we use the numerical experiments to verify the feasibility and effectiveness of the ROEFDI scheme.

A Reduced-Order Extrapolating Finite Difference Iterative Scheme for 2D Generalized Nonlinear Sine-Gordon Equation

In this study, a reduced-order extrapolating finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the two-dimensional (2D) generalized nonlinear Sine-Gordon equation is built via the proper orthogonal decomposition. The stability and convergence of the ROEFDI solutions are analyzed. And the feasibility and effectiveness of the ROEFDI scheme are verified by numerical simulations. This means that the ROEFDI scheme is effective and feasible for finding the numerical solutions of the 2D generalized nonlinear Sine-Gordon equation.

Two-phase natural convection dusty nanofluid flow

An analysis is performed to study the two-phase natural convection flow of nanofluid along a vertical wavy surface. The model includes equations expressing conservation of total mass, momentum and thermal energy for two-phase nanofluid. Primitive variable formulations (PVF) are used to transform the dimensionless boundary layer equations into a convenient coordinate system and the resulting equations are integrated numerically via implicit finite difference iterative scheme. The effect of controlling parameters on the dimensionless quantities such as skin friction coefficient, rate of heat transfer and rate of mass transfer is explored. It is concluded from the present analysis, that the diffusivity ratio parameter, NA and particle-density increment number, NB have pronounced influence on the reduction of heat transfer rate.

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

This study develops an optimized finite difference iterative (OFDI) scheme for the two-dimensional (2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition (POD) method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.

Double Diffusive Natural Convection Flow Over a Wavy Surface Situated in a Non-absorbing Medium

This analysis discuss the influence of heat and mass transfer on natural convection boundary layer flow of thermally radiating wavy surface. To make the surface radiating; Stephan-Boltzmann law is used in the boundary conditions. Therefore, convection and thermal radiation interact simultaneously with the aid of Stephan-Boltzmann law and ultimately producing highly nonlinear boundary conditions. The governing coupled boundary layer equations are switched into suitable form by using primitive variable formulations over which two-point iterative finite difference scheme is applied to obtain the unknown quantities. Physical quantities like wall shear stress, rate of heat transfer and rate of mass transfer are expressed graphically by varying significant emerging parameters: surface radiation (R), radiative length parameter (ξ), buoyancy ratio parameter (N) and amplitude of the wavy surface (α). Comparison of numerical results is also done in tabular form with the earlier study of Siddiqa et al. (2013) in order to validate the results.

Numerical solutions of nanofluid bioconvection due to gyrotactic microorganisms along a vertical wavy cone

An analysis is performed to study the bioconvection flow with heat and mass transfer of nanofluid containing gyrotactic microorganisms along a vertical wavy cone. The model includes equations expressing conservation of mass, momentum, thermal energy, nanoparticles and microorganisms. Primitive variable formulations (PVF) are used to transform the dimensionless boundary layer equations into a convenient coordinate system. The equations obtained from PVF are integrated numerically via an implicit finite difference iterative scheme. The effect of the controlling parameters on the dimensionless quantities such as local Nusselt number, local Sherwood number and local density number of motile microorganisms are explored. The numerical results show that the amplitude of the wavy surface of the cone and half cone angle has pronounced effect on the heat transfer coefficient, mass transfer coefficient and density number of the microorganisms coefficient.