Order Compact Finite Difference Scheme Research Articles

A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.

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.

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

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.

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.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

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.

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.

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.

Transient thermogravitational convection for magneto hybrid nanofluid in a deep cavity with multiple isothermal source-sink pairs

An analysis of fluid flow and thermal transfer has been encountered numerically for natural convection of Ag–MgO (50%–50%) water hybrid nanoliquid in a vertical enclosure permeated by a uniform magnetic field and different sets of heat source-sink pairs. The control equations (Navier-Stokes equations) constituting streamfunction (ψ) – vorticity (ζ) formulations with energy equation are solved by adopting a higher order compact finite difference scheme. Furthermore, the consequence of emerging parameters affecting fluid flows and heat transfer, namely, Hartmann number (0 ≤ Ha ≤ 60), hybrid nanoparticles volume fraction (0 ≤ φhnp ≤ 0.02), Rayleigh number (103 ≤ Ra ≤ 106), different cases (Case-I-III), aspect ratios (AR = 0.5, 1, 2, 4) and internal heat generation or absorption (−2 ≤ Q ≤ 2) are analyzed generously. Experimentally established correlations for thermal conductivity and dynamic viscosity have been used throughout the study. Obtained results ascertain that the arrangement of heat sources and heat sinks put major impact on fluid flow and heat transfer. The presence of magnetic field causes a reduction in flow strength which leads to decrease in thermal transmission and the maximum value of stream function. Moreover, Ag–MgO hybrid nanoparticles play indispensable role to levitate the thermal transmission compared to other parameters.

Sixth order compact finite difference schemes for Poisson interface problems with singular sources

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem −∇2u=f in Ω∖Γ with Dirichlet boundary condition such that f is smooth in Ω∖Γ and the jump functions [u] and [∇u⋅n→] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of −∇2u=f+gδΓ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇u⋅n→ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω+ and Ω−. The coefficient matrix A in the resulting linear system Ax=b, following from the proposed scheme, is independent of any source term f, jump condition gδΓ, interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω+ or Ω−. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

Fourth order compact FD methods for convection diffusion equations with variable coefficients

Fourth order finite difference methods combined with an integrating factor strategy for steady convection and diffusion partial differential equations with variable coefficients in both 2D and 3D are proposed using uniform Cartesian grids. An integrating factor strategy is applied to transform the convection and diffusion PDE to a self-adjoint form. Then, a fourth order finite difference method is obtained through a second order scheme followed by the Richardson extrapolation. Another approach is a direct fourth order compact finite difference scheme. The developed integrating factor strategy provides an efficient way for dealing with large convection coefficients. Several numerical examples are presented to demonstrate the convergence order and compare the two fourth order methods.

High Order Compact Block-Centered Finite Difference Schemes for Elliptic and Parabolic Problems

Based on the combination of block-centered and compact difference methods, fourth order compact block-centered finite difference schemes for the numerical solutions of one-dimensional and two-dimensional elliptic and parabolic problems with variable coefficients are derived and analyzed. Stability and optimal fourth-order error estimates are proved for both the solution and flux. Numerical experiments for model problems are presented to confirm the theoretical results and superior performance of the proposed schemes.

An efficient higher order compact finite difference schemes for linear hyperbolic partial differential equations

To approximate the partial derivative of Linear Hyperbolic Partial Differential Equations (LPHDE), a system of compact schemes used at non-boundary nodes. For the spatial discretization of the advection term, instead of using the upwind schemes, central difference based compact 4th order and conventional 2nd order schemes are experimented. The main aim of the numerical experiments carried out is, to assess the ability of the compact 4th order scheme in capturing the convection process, in comparison with the conventional 2nd order scheme.

Large eddy simulation of compressible round jets with coflow

Large eddy simulations of subsonic round jets are carried out using high order compact finite difference scheme and an explicit filtering based approximate deconvolution method. The jets have a Mach number of 0.9 and Reynolds number of 4.5×105 based on jet diameter and centerline velocity at inflow. Results obtained for the mean flow and turbulence intensities agree well with those in existing literature. We also study the effects of co-flow velocity ratio on the flow physics. Increase in potential core length and decrease in spreading rate of jet is observed in the presence of co-flow. The effects of co-flow velocity ratio on the axial Reynolds stress and turbulent kinetic energy budgets are also presented. It is observed that increasing co-flow velocity ratio leads to reduction in the turbulence intensities and near-field sound levels.

Theoretical understanding of unsteady flow separation for shear flow past three square cylinders in vee shape using structural bifurcation analysis

The unsteady flow separation of two-dimensional (2-D) incompressible shear flow past three identical square cylinders arranged in vee shape is studied in this paper, using theoretical structural bifurcation analysis based on topological equivalence. Through this analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with secondary and tertiary vortices on all cylinders are studied. The existence of saddle points is also studied during primary flow separation. Different gap ratios between the downstream cylinders, s/d = 0.6–3.0 (where s is the gap between cylinders, d is the length of cylinder side) with fixed gap 2d between upstream and downstream cylinders for different shear parameter (K) values ranging from \(K=0.0\) to 0.4 are considered at Reynolds number (Re) 100. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, phase diagram, lift and drag coefficients are studied to confirm the theoretical results. Computations are carried out by using higher order compact finite difference scheme. Present study mainly investigates the effect of K and gap ratio on unsteady flow separation and vortex-shedding phenomenon. All the computed results very efficiently and very accurately reproduce the complex flow phenomenon. Through this study, many noticeable and interesting results are reported for the first time for this problem.

A Structural Bifurcation Analysis of Flow Phenomenon for Shear Flow Past an Inclined Square Cylinder: Application to 2D Unsteady Separation

A structural bifurcation analysis of an incompressible two-dimensional (2D) shear flow past an inclined square cylinder by considering topological properties of flow is done in this paper. We have shown how the flow separation leads to complex structure at a time from a point by using this analysis. The streamfunction–vorticity ($$\Psi $$−$$\zeta $$) formulation of Navier–Stokes (N–S) equations in Cartesian coordinates is solved using a higher order compact (HOC) finite difference scheme. Through this analysis, we capture the exact locations of first and second bifurcation points with appropriate non-dimensional time of their occurrences for initial stages as well as fully developed flow. The flow field is mainly influenced by Reynolds number, Re, and shear rate, $$\kappa $$. It is shown that the first and second bifurcations developed within a very small time difference from the upper and lower downstream edges of the cylinder up to $$\kappa $$ = 0.1. Numerical simulations are carried out for Re = 100, 185 with $$\kappa $$ values range from 0 to 0.4. The purpose of the present study is to elaborate on the influence of shear parameter on flow properties. The temporal behavior of vortex formation and relevant streakline patterns are scrutinized for all parameter values. Occurrence of multiple separations is demonstrated in detail by varying $$\kappa $$ for both initial and fully developed flows. Comparisons with previous results in the literature clearly verify the accuracy and validity of the present work.

Effect of straining on spiral wave dynamics in excitable media

The present article envisages to explore the dynamics of stable rotating spiral waves in excitable media under the effect of shear strain through the Oregonator model. We employ an existing unconditionally stable, implicit high order compact finite difference scheme to discretize the highly nonlinear system governing the spiral wave patterns. In the process, we also study the effect of reaction parameters on the dynamics of spiral waves in the absence of straining, which results in the transition from the stable (periodic) rotation to compound (non-periodic) rotation of spiral waves. Our study reveals the existence of three regimes with a new time range, which is different from the earlier ones reported in literature; they characterize the extremely complex behaviour of the patterns under the effect of straining. The aforesaid conclusion is arrived at by carrying out a comprehensive analysis of the power spectrum of the tip motion. Besides establishing the stability of simulated rotating waves, the accuracy of the computed solutions has been ascertained through a convergence analysis and mesh independence study.