Standard Finite Difference Research Articles

Solving the Small Disturbance Equation and Numerical Method Optimization in Subsonic and Supersonic Flows

This paper addresses the solution of the Small Disturbance Equation (SDE) under subsonic and supersonic flow conditions using Successive Over-Relaxation (SOR) techniques. The study implements and validates numerical methods tailored to the unique dynamics of these regimes by setting precise boundary conditions, adjusting computational stencils, and managing the Mach number ( M∞ ) across the solution domain. A detailed comparison of streamline patterns between the regimes illustrates the efficacy of the applied numerical strategies. In the subsonic domain, the flow demonstrates uniform and smooth characteristics, conducive to standard SOR techniques, as evidenced by the rapid decline in residual errors, confirming the method’s efficiency for accurate solutions. Conversely, the supersonic regime presents increased complexity where standard finite difference methods encounter notable challenges, necessitating more sophisticated approaches to capture the intricate flow behaviors effectively. Conversely, in the supersonic regime, where the flow behavior exhibits more complex characteristics, the standard finite difference method faces challenges.

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

The GFMxP and the basic extrapolation of the ghost values to solve the Poisson equation for discontinuous functions

In a recent paper, a novel coding of the Ghost Fluid Method for the variable coefficient Poisson equation with discontinuous functions (named GFMxP) was proposed. A lot of numerical tests, with all the required quantities available in a analytic form, were used to demonstrate the ability of the new procedure in modeling a sharp interface and to check the accuracy order of the solutions. In practical applications, however, the real difficulty stands in the estimation of the so-called “ghost values”, that is the values at points where the function is not only unknown, but even not defined. These values allow to compute the corrective terms enabling the use of standard finite difference formulas in presence of a singularity and/or a discontinuity, and can be only determined through some extrapolation procedure, whose truthfulness is essential to achieve a reliable result. The paper deals with such a basic issue, by testing different numerical strategies and demonstrating the strict relationship between the order of the adopted fit-model, the order of the solving scheme for the Poisson equation and the accuracy of the final solution.

Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

In this work, we present a conservative Allen–Cahn (CAC) equation and investigate its unconditionally maximum principle-preserving linear numerical scheme. The operator splitting strategy is adopted to split the CAC model into a conventional AC equation and a mass correction equation. The standard finite difference method is used to discretize the equations in space. In the first step, the temporal discretization of the AC equation is performed by using the energy factorization technique. The discrete version of the maximum principle-preserving property for the AC equation is unconditionally satisfied. In the second step, we apply mass correction by using an explicit Euler-type approach. Without the constraint of time step, we estimate that the absolute value of the updated solution is bounded by 1. The unique solvability is analytically proved. In each time step, the proposed method is easy to implement because we only need to solve a linear elliptic type equation and then correct the solution in an explicit manner. Various computational experiments in two-dimensional and three-dimensional spaces are performed to confirm the performance of the proposed method. Moreover, the experiments also indicate that the proposed model can be used to simulate two-phase incompressible fluid flows.

Solving a fractional diffusion PDE using some standard and nonstandard finite difference methods with conformable and Caputo operators

IntroductionFractional diffusion equations offer an effective means of describing transport phenomena exhibiting abnormal diffusion pat-terns, often eluding traditional diffusion models.MethodsWe construct four finite difference methods where fractional derivatives are approximated using either conformable or Caputo operators.ResultsStability of the proposed schemes is analyzed using von Neumann stability analysis, and conditions are established to preserve positivity. Consistency analysis is performed for all methods, and numerical results with fractional parameters (α) set to 0.75, 0.90, 0.95, and 1.0 are presented.DiscussionThe rate of convergence in time for the four methods is computed.

A three-dimensional immersed boundary method for accurate simulation of acoustic wavefields with complex surface topography

Abstract Irregular topography of the free surface significantly affects seismic wavefield modelling, especially when employing finite-difference methods on rectangular grids. These methods represent the free surface as discrete points, resulting in a boundary that resembles a ‘staircase’. This approximation inaccurately represents surface topography, introducing errors in surface reflection traveltimes and generating artificial diffractions in wavefield simulation. We introduce a stable three-dimensional immersed boundary method (3DIBM) employing Cartesian coordinates to address these challenges. The 3DIBM enables the simulation of acoustic waves in media with complex topography through standard finite difference, extending the two-dimensional immersed boundary approach to compute spatial coordinates for ghost and mirror points in a three-dimensional space. Wavefield values at these points are obtained by three-dimensional spatial iterative symmetric interpolation, specifically through the Kaiser-windowed sinc method. By implicitly implementing the free surface boundary condition in three dimensions, this method effectively reduces artificial diffractions and enhances the accuracy of reflection traveltime. The effectiveness and accuracy of 3DIBM are validated through numerical tests and pre-stack depth migration imaging with simulated data, demonstrating its superiority as a modelling engine for migration imaging and waveform inversion in three-dimensional land seismic analysis.

A computational approach for 2D elliptic singularly perturbed weakly coupled systems of convection–diffusion type with multiple scales and parameters in the diffusion and the convection terms

In this work, we consider the efficient resolution of a 2D elliptic singularly perturbed weakly coupled system of convection–diffusion type, which has small parameters at both the diffusion and the convection terms. We assume that the diffusion parameters can be distinct at each equation of the system, and also, they can have different orders of magnitude, but the convection parameter is the same at both equations of the system. Then, in general, overlapping regular or parabolic boundary layers can appear in the exact solution. The continuous problem is approximated by using the standard upwind finite difference scheme, which is constructed on a special piecewise uniform Shishkin mesh. Then, the numerical scheme is an almost first‐order uniformly convergent method with respect to all the perturbation parameters. Some numerical results, obtained with the numerical algorithm for one test problem, are presented, which show the good performance of the proposed numerical method and also corroborate in practice the theoretical results.

Analytical Properties and Numerical Preservation of an Age-Group Susceptible-Infected-Recovered Model: Application to the Diffusion of Information

Abstract This paper analyzes an age-group susceptible-infected-recovered (SIR) model. Theoretical results concerning the conservation of the total population, the positivity of the analytical solution, and the final size of the epidemic are derived. Since the model is a nonlinear system of ordinary differential equations (ODEs), a numerical approximation is considered, based on Standard and non-Standard Finite Difference methods, and on a Modified Patankar-Runge–Kutta (MPRK) method. The numerical preservation of the qualitative properties of the analytical solution is studied. The obtained results are applied to the diffusion of information in social networks, and the effectiveness of the different numerical approaches is shown through several numerical tests on real data.

Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity

Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

Adaptive finite differencing in high accuracy electronic structure calculations

A multi-order Adaptive Finite Differencing (AFD) method is developed for the kinetic energy operator in real-space, grid-based electronic structure codes. It uses atomic pseudo orbitals produced by the corresponding pseudopotential codes to optimize the standard finite difference (SFD) operators for improved precision. Results are presented for a variety of test systems and Bravais lattice types, including the well-known Δ test for 71 elements in the periodic table, the Mott insulator NiO, and borax decahydrate, which contains covalent, ionic, and hydrogen bonds. The tests show that an 8th-order AFD operator leads to the same average Δ value as that achieved by plane-wave codes and is typically far more accurate and has a much lower computational cost than a 12th-order SFD operator. The scalability of real-space electronic calculations is demonstrated for a 2016-atom NiO cell, for which the computational time decreases nearly linearly when scaled from 18 to 144 CPU-GPU nodes.

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

Some finite difference methods for solving linear fractional KdV equation

The time-fractional Korteweg de Vries equation can be viewed as a generalization of the classical KdV equation. The KdV equations can be applied in modeling tsunami propagation, coastal wave dynamics, and oceanic wave interactions. In this study, we construct two standard finite difference methods using finite difference methods with conformable and Caputo approximations to solve a time-fractional Korteweg-de Vries (KdV) equation. These two methods are named as FDMCA and FDMCO. FDMCA utilizes Caputo's derivative and a finite-forward difference approach for discretization, while FDMCO employs conformable discretization. To study the stability, we use the Von Neumann Stability Analysis for some fractional parameter values. We perform error analysis using L1 &amp;amp; L∞ norms and relative errors, and we present results through graphical representations and tables. Our obtained results demonstrate strong agreement between numerical and exact solutions when the fractional operator is close to 1.0 for both methods. Generally, this study enhances our comprehension of the capabilities and constraints of FDMCO and FDMCA when used to solve such types of partial differential equations laying some ground for further research.

Mathematical Modelling of Tuberculosis and Diabetes Co-Infection Using the Non- Standard Finite Difference Scheme

One of the major health challenge facing Africa and in particular, Kenya is the risk of Tuberculosis and Diabetes. To understand the dynamics of this, a nine compartmental model for tuberculosis-diabetes co-infection is formulated. The Non-standard finite difference Scheme (NSFD) of the model is formulated from the first-order ordinary differential equations (ode) to avoid full implicit schemes that are computationally expensive. The overly small step sizes in NSFD give the user autonomy in controlling the accuracy of the results, making it suitable for disease control applications. Numerical simulations with different step sizes of the NSFD for the TB-Diabetes model are carried out to find the optimal step size, h. A comparison of the best resultant numerical simulation based on optimal h in NSDF indicates NSFD gives better results when compared with the corresponding first-order ode. The phase-plane analysis revealed that the NSFD formulated for tuberculosis and diabetes co-infection is generally asymptotically stable. Future studies should consider formulating the proposed model with varied control parameters such as medication to compare the results with those from first-order ode.

Finite difference spectral collocation schemes for the solutions of boundary value problems

This article introduces novel numerical approaches utilizing both standard and nonstandard finite difference methods to solve one-dimensional Bratu's problems. Using the quasilinearization technique, the original problem is converted into a sequence of linear problems. Chebyshev polynomials are employed to approximate the second derivative of the function y(x), after which Sumudu transform is applied to obtain a new form of trial function. The obtained trial function is then substituted into a linearized and discretized Bratu's equations. We discuss the convergence of the schemes and compare the numerical outcomes to those derived using other relevant methods. We further modify one of the new schemes and apply it to solve boundary value problem with associated Robin conditions. The results show that the proposed schemes yield accurate approximations to the solutions of the problems considered.

A dispersion analysis of uniformly high order, interior and boundaries, mimetic finite difference solutions of wave propagation problems

A preliminary stability and dispersion study for wave propagation problems is developed for mimetic finite difference discretizations. The discretization framework corresponds to the fourth-order staggered-grid Castillo-Grone operators that offer a sextuple of free parameters. The parameter-dependent mimetic stencils allow problem discretization at domain boundaries and at the neighbor grid cells. For arbitrary parameter sets, these boundary and near-boundary mimetic stencils are lateral, and we here draw first steps on the parametric dependency of the stability and dispersion properties of such discretizations. As a reference, our analyses also present results based on Castillo-Grone parameters leading to mimetic operators of minimum bandwidth that have been previously applied in similar physical problems. The most interior parameter-dependent mimetic stencils exhibit a specific Toeplitz-like structure, which reduces to the standard central finite difference formula for staggered differentiation at grid interior. Thus, our results apply to the whole discretization grid. The study done for the 1-D problem could be applied to the discretization of a free surface boundary condition along an orthogonal gridline to this boundary.

Image trinarization using a partial differential equation: A novel approach to automatic sperm image analysis

Partial differential equations have recently garnered substantial attention as an image processing framework due to their extensibility, the ability to rigorously engineer and analyse the governing dynamics as well as the ease of implementation using numerical methods. This paper explores a novel approach to image trinarization with a concrete real-world application of classifying regions of sperm images used in the automatic analysis of sperm morphology. The proposed methodology engineers a diffusion equation with non-linear source term, exhibiting three steady-states. The model is implemented as an image processor using a standard finite difference method to illustrate the efficacy of the proposed approach. The performance of the proposed approach is benchmarked against standard image clustering/segmentation methods and shown to be highly effective.

Numerical Computation of Natural Convection of Nanofluid in an Open Wavy Porous Cavity Heated Partially

In this work, we emphasise the heat transfer and fluid flow due to buoyancy force in a wavy open porous cavity, placed horizontally having filled with porous media and, its top wall is being kept open, whereas the right cold wall is wavy and, its left vertical wall is heated partially, keeping all other walls at thermally insulated. The non-dimensional ψ − θ formulation of mass, momentum, and energy conservation laws for porous media are solved by the standard finite difference scheme for a wide range of pertinent parameters such as nanoparticle volume fraction (0.05 ≤ Φ ≤ 0.2), Rayleigh-Darcy number (10 ≤ Ra ≤ 103), length of heat source (0.25 ≤ ε ≤ 1), and parameters controlling waviness of right wall (1 ≤ N ≤ 5) and amplitude (0.05 ≤ a ≤ 0.25). The simulated results are presented in the form of streamlines and isotherms; global and local Nusselt numbers are computed. Obtained results are analyzed and it is observed that the convection process is augmented at the presence of nanoparticle for low Ra but decreases at high Ra for all pertinent parameters; also, the wall waviness augments convection low Ra.

A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

This work proposes a boundary integral formulation based Cartesian grid method for the nonlinear Poisson-Boltzmann (PB) interface problem in biophysics. The method (a) does not have the limitation associated with the standard finite difference method for complex interfaces, (b) avoids the generation of any unstructured volume or surface grids needed by the finite element method, and (c) solves the nonlinear and variable coefficient PDE in the framework of boundary integral equations. The method solves the nonlinear PB equation with the Newton iterative method. It first reformulates the linearized PB equation in each Newton iteration as a Fredholm system of two boundary integral (BI) equations of the second kind, which is well-conditioned, and then solves the BI system with a Krylov subspace method. The evaluation of boundary and volume integrals involved in the solution of the BI system is done with a kernel-free method, which does not need to know Green's function associated with the variable coefficient PB equation. The kernel-free method evaluates volume and boundary integrals by solving equivalent simple interface problems on Cartesian grids with either a fast Fourier transform based Poisson solver or a geometric multigrid preconditioned conjugate gradient solver, either of which involves computational work essentially linearly proportional to the number of nodes on the Cartesian grid. Numerical examples in both two and three space dimensions are presented to demonstrate the efficiency and accuracy of the proposed numerical method.