Order Compact Scheme Research Articles

Combined high order compact schemes for non-self-adjoint nonlinear Schrödinger equations

Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives ux‾, uxx in the NSANLSE. If one uses classical high order compact schemes to approximate uxx and ux‾ separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves uxjn and uxxjn simultaneously in terms of uj. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.

A novel transformation free nonuniform higher order compact finite difference scheme for solving incompressible flows on circular geometries

This paper presents a newly developed higher order compact scheme that is designed on a polar grid by using an implicit form of first order derivatives on nonuniform grids. These derivatives are formulated compactly and implicitly with a relationship of the coefficients in terms of the unknown variables. The proposed scheme is free from transformation technique and third order accurate in space. The objective is to solve Stokes equations and Navier–Stokes equations on curvilinear grids using the polar nature of the coordinate system. Our newly developed scheme is used to solve several problems namely, a problem having an analytical solution, Stokes flow with different orientations of the lids for the half filled annular and wedge cavity, the lid driven polar cavity flow and flow past an impulsively started circular cylinder. Our newly developed scheme is used to analyze the flow structures for the flow governed by different physical control parameters: the cavity radius ratio, the cavity angle and the ratio of the upper and lower lid speeds with rotating coaxial cylinders and Reynolds number for the impulsively started circular cylinder. The Stokes equations and the Navier–Stokes equations are efficiently solved with Dirichlet as well as Neumann boundary conditions. The efficacy and robustness of our proposed scheme are shown through its applicability in all the complex fluid flow problems.

Flow dynamics and heat transfer in the wake of an rotary oscillating circular cylinder with an isothermal control plate

This research explores the flow dynamics and heat transfer within the wake of a rotary oscillating circular cylinder in the presence of an isothermal control plate. Two-dimensional, unsteady, viscous and laminar flow of a Newtonian fluid is considered using the Higher Order Compact Scheme (HOC) to discretize the governing equations and the Bi-Conjugate Gradient Stabilized method to solve the resulting system. Simulations are performed for various gap ratios, maximum angular velocities, and frequency ratios of oscillation at Prandtl number 0.7 and Reynolds number 150 using an in-house code. Results show a significant increase in heat transmission for d / R 0 = 0.5 and f / f 0 = 0.5 for all αm . Drag and lift coefficients are also analyzed, with the maximum peak of the drag coefficient decreasing by 9.88% for d / R 0 = 3 . This research offers valuable insights for advancing and optimizing aerodynamic forces and heat transfer processes, particularly in the field of fluid dynamics. The study focuses on enhancing the efficiency of heat transfer mechanisms around rotary oscillating circular cylinders, contributing to the development of cutting-edge technologies in aerodynamics and thermal management.

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

High order compact augmented methods for Stokes equations with different boundary conditions

This paper is devoted to fourth order compact schemes and fast algorithms for solving stationary Stokes equations with different boundary conditions numerically. One of the main ideas is to decouple the Stokes equations into three Poisson equations for the pressure and the velocity via the pressure Poisson equation (PPE). The augmented strategy is utilized to provide numerical boundary conditions for the pressure. Different velocity boundary conditions require different interpolation strategies for the augmented methods. The augmented variable is solved by the GMRES method. A new simple and efficient preconditioning strategy has also been developed to accelerate the convergence of the GMRES iteration. Numerical examples presented in this paper confirmed the designed convergence order and the efficiency of the new methods.

Optimizing thermosolutal and hydrothermal performance of radiative hybrid ferrofluid and entropy generation in a wavy porous enclosure

This research conducts a comprehensive investigation to enhance the thermosolutal and hydrothermal efficiency of a radiative Fe3O4–Cu–H2O hybrid ferrofluid within a wavy porous enclosure with the enlightenment for heat transfer and material science. The primary focus is on optimizing the thermosolutal performance in this irregular enclosure in which the lower boundary undergoes non-uniform heating and concentration, and the curved lateral boundaries keep cooling, lower concentration. Simultaneously, the upper flat boundary remains adiabatic. Using an efficient Higher Order Compact (HOC) scheme, we solve the governing streamfunction (ψ)-vorticity (ζ) form with the incorporation of energy and species transport equations of the coupled Navier–Stokes equations. Our results are robust and validated against established experimental and numerical data sources. The investigation delves into the impacts of well-defined parameters such as Rayleigh number (Ra), Hartmann number (Ha), Darcy number (Da), Buoyancy ratio (N), Lewis number (Le), Radiation parameter (Rd), shape factor of nanoparticles (m) and solid volume fraction (ϕhnp) of the hybrid ferrofluid. Key findings underscore the consistent enhancement of both average Nusselt number (Nu¯) and average Sherwood number (Sh¯) with higher Buoyancy Ratio Numbers (N). Blades-shaped nanoparticles (m=8.6) demonstrate superior performance in both unitary and hybrid ferrofluid systems. Optimal conditions, identified at ϕhnp=0.04 and Ra=104 by the Bejan number (Be), underscore the positive impact of higher Buoyancy Ratio Numbers (N), resulting in a 25.41% increase in Nu¯ and an impressive 43.81% boost in Sh¯ observed from the shifting of N=0 to N=10.

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

<abstract><p>In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.</p></abstract>

Accurate derivatives approximations and applications to some elliptic PDEs using HOC methods

For many application problems that are modeled by partial differential equations (PDEs), not only it is important to obtain accurate approximations to the solutions, but also accurate approximations to the derivatives of the solutions. In this study, some new high order compact (HOC) finite difference schemes are derived to approximate the first and second derivatives of the solution to some elliptic PDEs using the numerical solution obtained from a HOC scheme applied to the same PDE. Convergence analysis for the computed derivatives is also presented to show that the order of the convergence is the same as that of the solution. The new HOC schemes for computing partial derivatives at both interior and boundary grid points take into account of the partial differential equations including the source term and/or the boundary conditions (Dirichlet, Neumann, or Robin). Fourth order accurate compact finite difference formulas with pre-computed coefficients and weights are developed for Poisson/Helmholtz PDEs, and code generated coefficients for diffusion-advection equations with constant coefficients. One important application is a new fourth-order compact scheme for solving incompressible Stokes equations with periodic boundary conditions.

Thermogravitational magnetonanofluid flow in a square cavity with isotropic heat flow field and viscous dissipation: A compact approach

In this paper, thermogravitational flow in a square enclosure filled with Cu-water nanofluid which is influenced by inclined magnetic induction, has been mathematically modelled. Isotropic heat flow field and viscous dissipation effects are introduced in the modelling as well. The vertical walls of the domain are subjected to finite temperature gradient while the horizontal walls are thermally insulated. The control equations subject to the flow physics of an incompressible, viscous and electrically conducting fluid are the Navier-Stokes equations along with energy equation. These governing equations are solved numerically by proposing a fourth order compact scheme. The influences of diverse effective parameters including Hartmann number, viscous dissipation parameter (Eckert number) and heat generation or absorption parameter on characteristic features of heat transfer and fluid flow with different concentration distributions of nanoparticles are analyzed and addressed in detail. Results show that the presence of viscous dissipation retards the thermal diffusion process. The average Nusselt number values are decreased with the increasing values of Eckert number. In addition, with the increasing values of Hartmann number, the thermal transport is decreased in spite of incorporation of isotropic heat flow.

A 3D frequency-domain electromagnetic solver employing a high order compact finite-difference scheme

Three-dimensional (3D) electromagnetic (EM) modeling is an important tool for geophysical applications. In EM induction methods the modified Helmholtz equation is often used to describe scattered or residual electric fields in three dimensions. Throughout this paper, a high order compact finite difference scheme for the solution of that equation for vertical magnetic dipole source (VMD) is presented. The approximation of the residual electric field intensity using a fourth order compact finite difference (FD) discretizer is achieved with the solution of a block linear system, the coefficient matrices are large and sparse with a particular structure, implying the application of a matrix-free Krylov subspace method for an efficient numerical solution. The proposed solver is being examined using a number of test problems in uniform and non-uniform grid spacing where numerical and analytical solutions for the homogeneous half-space cases are being compared.

High Order Compact Schemes for Flux Type BCs

High Order Compact Schemes for Flux Type BCs

Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.

Effect of arc‐shaped vertical control plate on heat and mass transfer in uniform flow past an isothermally heated circular cylinder

AbstractThe current work investigates the effect of an arc‐shaped vertical control plate on the heat and mass transfer in uniform flow past an isothermally heated circular cylinder. The control plate is positioned downstream at various distances from the circular cylinder's surface. The governing equations are discretized by the higher order compact (HOC) finite difference scheme, and then this system of discretized equations is solved by using a bi‐conjugate gradient stabilized iterative method for Prandtl number , Reynolds number . To investigate the effect of an arc‐shaped control plate on heat and mass transfer, we consider a range of nondimensional distances between the circular cylinder and the control plate, , where is the control plate's distance and is the cylinder's radius. The exact timing and location of the bifurcation points are calculated by using topological aspect‐based structural bifurcation analysis. Significant effects of various locations of the control plate on periodic wake and heat transfer are observed. It is found that the increasing distance of the control plate from the cylinder delays the occurrence of the structural bifurcation and shifts the bifurcation points upwards in the upper half and downwards in the lower half of the cylinder. Our study shows that the specific location of the control plate can fully suppress the vortex shedding. Time‐averaged total Nusselt number can be drastically reduced by increasing the distance between the control plate and the cylinder. With proper positioning, the vertical control plate can lessen the time‐averaged drag force by up to when compared to a cylinder without a control plate. Overall, this work presents many new phenomena that have not been reported before.

Magneto-thermogravitational convection for hybrid nanofluid in a novel shaped enclosure

The proposed work exhibits a numerical simulation of steady thermogravitational convection of Cu–Al2O3(50%−50%) water hybrid magneto-nanofluid in a novel shaped enclosure regulated by magnetic field dependent (MFD) viscosity in presence of flush mounted variable heaters. Various cases (Case-I to Case-IV) have been considered depending on the location of the heaters on the bottom and left boundaries. Pertaining to this analysis, the streamfunction (ψ)- vorticity (ζ) form including the energy transport equation of coupled partial differential equations are evaluated with the help of an efficient higher order compact finite difference scheme. Results are exhibited in terms of streamlines, isotherms and Nusselt numbers to describe the transport phenomena obtained for different physical parametric values to this simulation. The influence of the physical parameters such as, Rayleigh number (104≤Ra≤106), Hartmann number (0≤Ha≤60), hybrid nanoparticles volume fraction (0≤ϕhnp≤0.04), orientation of magnetic field (00≤γ≤600) and magnetic number (0≤δ0≤1) on the hybrid nanofluid flow and energy transfer are analyzed. Obtained results indicate that the combinations of the heaters location and the shaped of the geometry have major impact on all the transport phenomena. Moreover, Cu–Al2O3 hybrid nanoparticles play an essential role to levitate the thermal transmission compared to other physical parameters. An addition of hybrid nanoparticles leads to an increase in the average Nusselt number up to 10.8% in Case-I, 12% in Case-II, 11.66% in Case-III and 11.65% in Case-IV. The novelty of the present work is to examine the rheological properties of hybrid nanoliquid in a novel shaped cavity subjected to magnetic field dependent viscosity (MFDV) along with flush mounted variable heaters.

Influence of a circular obstacle on the dynamics of stable spiral waves with straining

The current study envisages to investigate numerically, probably for the first time, the combined effect of a circular obstacle and medium motion on the dynamics of a stable rotating spiral wave. A recently reconstructed spatially fourth and temporally second order accurate, implicit, unconditionally stable high order compact scheme has been employed to carry out simulations of the Oregonator model of excitable media. Apart from studying the effect of the stoichiometric parameter, we provide detailed comparison between the dynamics of spiral waves with and without the circular obstacles in the presence of straining effect. In the process, we also inspect the dynamics of rigidly rotating spiral waves without straining effect in presence of the circular obstacle. The presence of the obstacle was seen to trigger transition to non-periodic motion for a much lower strain rate.

A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let Γ be a smooth curve inside a rectangular region Ω. In this paper, we consider the elliptic interface problem −∇·(a∇u)=f in Ω∖Γ with Dirichlet boundary conditions, where the coefficient a and the source term f are smooth in Ω∖Γ and the two jump condition functions [u] and [a∇u·n→] across Γ are smooth along the interface Γ. To solve such elliptic interface problems, we propose a high order compact 9-point finite difference scheme and a high order local calculation for numerically computing the solution u and its gradient ∇u respectively on uniform Cartesian grids without changing coordinates into local coordinates. We numerically verify the sign conditions of our proposed compact finite difference scheme and prove the convergence rate by the discrete maximum principle. Our numerical experiments confirm the fourth order accuracy for computing the solution u in both l2 and l∞ norms of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

Numerical solution of the two-dimensional regularized long-wave equation with a conservative linearized high order finite difference scheme

In this article, a finite difference scheme is investigated to solve a two-dimensional regularized long-wave equation. The original equation is deformed into another formulation. The fourth order compact difference scheme and the fourth order Padé scheme are used for the treatment of the second and first derivatives of the deformation equation in space direction, and the fourth order backward difference formula is used for the discretization of the time derivative. The nonlinear term in the proposed scheme is linearized by Taylor series expansion method. The new scheme is shown to be of the fourth order accuracy in both temporal and spatial dimensions. The existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the new scheme is shown to be conservative and unconditionally stable. Finally, numerical experiments are provided to verify our theoretical analysis results of the new scheme.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

Heat Transfer Past a Rotationally Oscillating Circular Cylinder in Linear Shear Flow

Abstract This study investigates the unsteady, two-dimensional flow and heat transfer past a rotationally oscillating circular cylinder in linear shear flow. A higher order compact (HOC) finite difference scheme is used to solve the governing Navier–Stokes equations coupled with the energy equation on a nonuniform grid in polar coordinates. The hydrodynamic and thermal features of the flow are mainly influenced by the shear rate (K), Reynolds number (Re), Prandtl number (Pr), and the cylinder oscillation parameters, i.e., oscillation amplitude (αm), the frequency ratio (fr). The simulations are performed for Re=100,Pr=0.5−1.0, 0.0≤K≤0.15, and 0.5≤αm≤2.0. The numerical scheme is validated with the existing literature studies. Partial and full vortex suppression is observed for certain values of shear parameter K. The connection between heat transfer and vortex shedding phenomenon is examined where a pronounced increase in the heat transfer is observed for certain values of oscillation parameter, relative to the nonshear flow case.

On the efficiency of 5(4) RK-embedded pairs with high order compact scheme and Robin boundary condition for options valuation

When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare the performance of our present method with several 5(4) Runge–Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.