Stable Finite Difference Scheme Research Articles

Finite difference approach in porous media transport modeling for magnetohydrodynamic unsteady flow over a vertical plate

Purpose – The purpose of this paper is to examine the effects of Darcian drag force and radiation-conduction on unsteady two-dimensional magnetohydrodynamic flow of viscous, electrically conducting and Newtonian fluid over a vertical plate adjacent to a Darcian regime in presence of thermal radiation and transversal magnetic field. A well-tested, numerically stable Crank-Nicolson finite-difference procedure is employed for the conservation equations. Excellent agreement is obtained for numerical solutions with previously published work. Design/methodology/approach – In this investigation, an efficient, accurate, extensively validated and unconditionally stable finite-difference scheme based on the Crank-Nicolson model is developed to solve the governing coupled, non-linear partial differential equations. The accuracy and effectiveness of the method are demonstrated. Findings – Different numerical results are obtained and presented graphically to explain the effect of various physical parameters on the velocity and temperature profiles, local, as well as average, skin friction and Nusselt number. It is found that, with a rise in Darcian drag force, flow velocity and temperature are reduced, but increased for all times. Both average and local skin frictions are reduced considerably with an increase in Darcian drag force, but reversed behavior is observed for the local Nusselt number. Increasing the thermal radiation effects accelerated the flow velocity as well as the fluid temperature and wall local skin friction in a saturated porous medium, but effectively reduced the local Nusselt number and average Nusselt number at the wall. Comparison with previously published works in the limits shows excellent agreement. Research limitations/implications – The analysis is valid for unsteady, two-dimensional laminar flow of an optically thick no-gray gas, electrically conducting, and Newtonian fluid past an isothermal vertical surface adjacent to the Darcian regime with variable surface temperature. An extension to three-dimensional flow case is left for future work. Practical implications – Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. It is well known that the effect of thermal radiation is important in space technology and high temperature processes. Thermal radiation also plays an important role in controlling heat transfer process in polymer processing industry.

On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions

A new explicit conditionally consistent finite difference scheme for one-dimensional third-order linear pseudoparabolic equation with nonlocal conditions is constructed. The stability of the finite difference scheme is investigated by analysing a nonlinear eigenvalue problem. The stability conditions are stated and stability regions are described. Some numerical experiments are presented in order to validate theoretical results.

An energy-stable finite-difference scheme for the binary fluid-surfactant system

We present an unconditionally energy stable finite-difference scheme for the binary fluid-surfactant system. The proposed method is based on the convex splitting of the energy functional with two variables. Here are two distinct features: (i) the convex splitting energy method is applied to energy functional with two variables, and (ii) the stability issue is related to the decay of the corresponding energy. The full discrete scheme leads to a decoupled system including a linear sub-system and a nonlinear sub-system. Algebraic multigrid and Newton-multigrid methods are adopted to solve the linear and nonlinear systems, respectively. Numerical experiments are shown to verify the stability of such a scheme.

Thermal radiation effect on an unsteady MHD natural convection Couette flow with Heat and mass transfer

The numerical solution of natural convection in unsteady hydromagneticCouette flow of a viscous incompressible electrically conducting fluid between two vertical parallel plates in the presence of thermal radiation is obtained here. The fluid is considered to be a gray, absorbing – emitting but non – scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The dimensionless governing coupled, non – linear boundary layer partial differential equations are solved by an efficient, accurate, and extensively validated and unconditionally stable finite difference scheme of the Crank – Nicolson method. Computations are performed for a wide range of the governing flow parameters, viz., the thermal Grashof number, SolutalGrashof number, Magnetic field parameter (Hartmann number), Prandtl number, Thermal radiation parameter and Schmidt number. The effects of these flow parameters on the velocity and temperatureare shown graphically. Finally, the effects of various parameters on the on the skin – friction coefficient and Rate of heat and mass transferat the wall are prepared with various values of the parameters.These findings are in quantitative agreement with earlier reported studies.

MATHEMATICAL MODELING FOR POROUS MEDIA TRANSPORT IN NEWTONIAN RADIATING/CHEMICALLY REACTING FLUID OVER AN IMPULSIVELY-STARTED VERTICAL PLATE: A FINITE DIFFERENCE APPROACH

Theoretical investigation is presented here for the model of unsteady MHD thermal convection flow of a viscous incompressible absorbingemitting optically thin gray gas along an impulsivelystarted semi-infinite vertical plate adjacent to the Darcian porous regime in the presence of a first order chemical reaction and significant thermal radiation effects. The conservation equations are nondimensionalized and are solved by an accurate and unconditionally stable implicit finite difference scheme of Crank-Nicholson type. The flow is found to be accelerated with increasing porosity parameter (K), whereas the temperature and concentration distributions are reduced in the Darcian regime. Flow velocity and Temperature are found to be depressed with progression of thermal radiation (Ra) contribution, but enhanced the concentration distribution. Applications of the model arise in solar energy collector analysis, magneto-fluid dynamics and industrial materials processing.

Stability of finite difference schemes for generalized von Foerster equations with renewal

We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \(l^1\) and \(l^\infty\) norms.

Effects of Chemical Reactions on Unsteady Free Convective and Mass Transfer Flow from a Vertical Cone with Heat Generation/Absorption in the Presence of VWT/VWC

A mathematical model for the effects of chemical reaction and heat generation/absorption on unsteady laminar free convective flow with heat and mass transfer over an incompressible viscous fluid past a vertical permeable cone with nonuniform surface temperatureTw'(x)=T∞'+axnand concentrationCw'(x)=C∞'+bxmis considered here. The dimensionless governing boundary layer equations of the flow that are transient, coupled, and nonlinear partial differential equations are solved by an efficient, accurate, and unconditionally stable finite difference scheme of Crank-Nicholson type. The velocity, temperature, and concentration profiles have been studied for various parameters, namely, chemical reaction parameterλ, the heat generation and absorption parameterΔ, Schmidt number Sc, Prandtl numberPr, buoyancy ratio parameterN, surface temperature power law exponentn, and surface concentration power law exponentm. The local as well as average skin friction, Nusselt number, and Sherwood number are discussed and analyzed graphically. The present results are compared with available results in open literature and are found to be in excellent agreement.

Unsteady Free Convection in a Walter’s-B Viscoelastic Flow past a Semi-Infinite Vertical Plate with Radiation and Chemical Reaction

A numerical solution for the free convective, unsteady, laminar convective heat and mass transfer in a viscoelastic fluid along a semi-infinite vertical plate with radiation and chemical reaction is presented. The Walters-B liquid model is employed to simulate medical creams and other rheological liquids encountered in biotechnology and chemical engineering. This rheological model introduces supplementary terms into the momentum conservation equation. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Prandtl number, viscoelasticity parameter, Schmidt number, radiation parameter, chemical reaction parameter and buoyancy parameters. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed graphically. It is observed that, when the viscoelasticity parameter increases, the velocity increases close to the plate surface. An increase in Schmidt number is observed to significantly decrease both velocity and concentration.

NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.

Analytical and Numerical Study for MHD Radiating Flow over an Infinite Vertical Surface Bounded by a Porous Medium in Presence of Chemical Reaction

The study of non-linear MHD flow with heat and mass transfer characteristics of an incompressible, viscous, electrically conducting and Newtonian fluid over a vertical oscillating porous plate embedded in a porous medium in presence of homogeneous chemical reaction of first order and thermal radiation effects have been analyzed. The fluid considered here is a gray, absorbing/emitting radiation, but a non-scattering medium. At time , the plate temperature and concentration levels near the plate raised linearly with time . The dimensionless governing coupled, non-linear boundary layer partial differential equations are solved by an efficient, accurate, and extensively validated and unconditionally stable finite difference scheme of the Crank-Nicolson type as well as by the Laplace Transform technique. An increase in porosity parameter is found to depress the fluid velocities and shear stress in the regime. Also it has been found that, when the conduction-radiation increased, the fluid velocities as well as temperature profiles were decreased. It has been found that, when the chemical reaction parameter increased, the fluid velocities as well as concentration profiles were decreased. Applications of the study arise in materials processing and solar energy collector systems.

Entropy/energy stable schemes for evolutionary dispersal models

In this paper we propose some entropy/energy stable finite difference schemes for the reaction–diffusion–advection equation arising in the evolution of biased dispersal of population dynamics. The peculiar feature of these active dispersal models is that the transient solution converges to the stable steady state when time goes to infinity. For the numerical method to capture the long-time pattern of persistence or extinction, we use the relative entropy when the resource potential is logarithmic, and explore the usual energy for other resource potentials. The present schemes are shown to satisfy three important properties of the continuous model for the population density: (i) positivity preserving, (ii) equilibrium preserving, and (iii) entropy or energy satisfying. These ensure that our schemes provide a satisfying long-time behavior, thus revealing the desired dispersal pattern. Moreover, we present several numerical results which confirm the second-order accuracy for various resource potentials and underline the efficiency to preserve the large time asymptotic.

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Nonlinear entropy stability is used to derive provably stable high-order finite difference operators including boundary closure stencils, for the compressible Navier–Stokes equations. A comparison technique is used to derive a new Entropy Stable Weighted Essentially Non-Oscillatory (SSWENO) finite difference method, appropriate for simulations of problems with shocks. Viscous terms are approximated using conservative, entropy stable, narrow-stencil finite difference operators. The efficacy of the new discrete operators is demonstrated using both smooth and discontinuous test cases.

Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation

In this paper we present two unconditionally energy stable finite difference schemes for the modified phase field crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic phase field crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ɛ-uniformly in the maximum norm at an O(N−1lnN + N0−1) rate, where N + 1 and N0 + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ−2lnδ−1 + δ0−1); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N−1lnN and δ0 = N0−1 are the accuracies of the discrete solution in x and t, respectively.

Higher Order Time-Domain Finite-Element Method for Microwave Device Modeling With Generalized Hexahedral Elements

A novel higher order and large-domain Galerkin-type finite-element method (FEM) is proposed for direct 3-D electromagnetic modeling in the time domain. The method is implemented in the time-domain finite-element method (TDFEM) analysis of multiport microwave waveguide devices with arbitrary metallic and dielectric discontinuities. It is based on the geometrical modeling using Lagrange-type interpolation generalized hexahedra of arbitrary geometrical-mapping orders, field expansion in terms of hierarchical curl-conforming 3-D polynomial vector basis functions of arbitrarily high field-approximation orders, time stepping with an implicit unconditionally stable finite-difference scheme invoking the Newmark-beta method, and mesh truncation introducing the waveguide port boundary condition. Numerical examples demonstrate excellent accuracy, efficiency, stability, convergence, and versatility of the presented method, and very effective large-domain TDFEM models of 3-D waveguide discontinuities using minimal numbers of large conformal finite elements and minimal numbers of unknowns. The results obtained by the higher order TDFEM are in an excellent agreement with indirect solutions obtained from the FEM analysis in the frequency domain (FD) in conjunction with the discrete Fourier transform and its inverse, as well as with measurements and with alternative full-wave numerical solutions in both time and FDs.

On the numerical solution of hyperbolic IBVP with high-order stable finite difference schemes

The abstract Cauchy problem for the hyperbolic equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented. The stability estimates for the solutions of these difference schemes are established. A finite difference method and some results of numerical experiments are presented in order to support theoretical statements. MSC:65J10, 65M12, 65N12, 35L30.

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $$\ell ^2$$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).

Flow and Heat Transfer of Casson Fluid from a horizontal Circular Cylinder with Partial Slip in non-Darcy porous Medium

In the Present study, the steady flow and heat transfer of Casson fluid from a permeable horizontal cylinder in the presence of slip condition in a non-Darcy porous medium is analyzed. The cylinder surface is maintained at a constant temperature. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite-difference scheme. Increasing the velocity slip parameter is found to decrease the velocity and boundary layer thickness and increases the temperature and the boundary layer thickness. The velocity decreases with the increase the non- Darcy parameter and is found to increase the temperature. The velocity increases with the increase the Casson fluid parameter and is found to decrease the temperature. The Skin-friction coefficient and the local Nusselt number are found to decrease with the increase in velocity and thermal slip parameters respectively

A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.

Transient Free Convective Flow over a Vertical Cone Embedded in a Thermally Stratified Medium

Unsteady natural convection flow of a viscous and incompressible flow over a vertical cone immersed in a stable thermally stratified medium is theoretically studied in this paper. The dimensionless coupled partial differential boundary layer equations are solved numerically using an efficient and unconditionally stable finite-difference scheme of Crank-Nicolson type. The effects of Prandtl number and stratification parameter on the velocity and temperature profiles on the flow and heat transfer characteristics have been determined and discussed in detail. The present results are compared with available results from the open literature and are found to be in good agreement. We notice that natural convection flow over a vertical surface embedded in porous media also occurs in several engineering problems such as those encountered in the design of pebble-bed nuclear reactors, catalytic reactors, and compact heat exchangers, in geothermal energy conversion, in the use of fibrous materials, in the thermal insulation of buildings, in the heat transfer from storage of agricultural products which generate heat as a result of metabolism, in petroleum reservoirs, in nuclear wastes, etc.