Second-order Finite Difference Research Articles

A robust, exponentially fitted higher-order numerical method for a two-parameter singularly perturbed boundary value problem

This study constructs a robust higher-order fitted operator finite difference method for a two-parameter singularly perturbed boundary value problem. The derivatives in the governing ordinary differential equation are substituted by second-order central finite difference approximations, after which the fitting parameter is introduced and determined. The resulting system of linear equations may then be solved using the Thomas method. The stability, consistency, and convergence of the current method have been thoroughly validated. To enhance accuracy and achieve a higher-order numerical solution, a post-processing technique was employed to upgrade the method from second-order to fourth-order convergence. Finally, three test examples were used to confirm the method's appropriateness. The numerical results demonstrate that the proposed technique is stable, consistent, and produces a higher-order numerical solution than the existing ones in the literature.

Numerical analysis of a second-order finite difference scheme for Riesz space-fractional Allen–Cahn equations

The goal of this paper is to present a second-order finite difference scheme for Allen–Cahn equations with Riesz fractional derivative. The discrete maximum bound principle, the maximum-norm error estimates, and the discrete energy stability of the proposed scheme are discussed. It is shown that the proposed scheme is unconditionally energy-stable for any nonnegative stabilization parameter. Two numerical experiments are performed to verify the theoretical results.

New accurate conservative finite difference schemes for 1-D and 2-D Schrödinger-Boussinesq Equations

In this paper, first-order and second-order accurate structure-preserving finite difference schemes are proposed for solving the Schrödinger- Boussinesq equations. The conservation of the discrete energy and mass of the present schemes are analytically proved. Numerical experiments are given to support the theoretical results. Numerical examples show the efficiency of the proposed scheme and the correction of the theoretical proofs

Higher-order approximations for space-fractional diffusion equation

Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyse the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order in the range . Numerical examples further confirm the convergence order and unconditional stability of the approximations, demonstrating their effectiveness in practice.

The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number

The mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (Rλ). This value is associated with the well-scaling dissipation spectrum in Kolmogorov units, where the maximum corresponds to the bottleneck peak. Even the scalar dissipation rate at the high Rλ considered in the present direct numerical simulations attains a constant value as Sc increases. In this scenario, the maximum of the scalar dissipation spectra reaches its peak within the bottleneck, starting at Sc>0.5. A qualitative explanation for the formation of the two bottlenecks is related to the blockage of energy transfer from large to small scales in the inertial ranges. Within the bottleneck, the self-similar, ribbon-like structures transition into the rod-like structures characteristic of the exponential decay range. Investigating the viscous dependence of the bottleneck’s amplitude may be aided by examining the evolution of a passive scalar. As Sc decreases, the scalar spectra undergo changes across the wave number k range. The bottleneck is dismantled, and at very low Sc values, the spectrum tends towards Batchelor’s theoretical prediction, diminishing proportionally to k−17/3. To comprehend the flow structures responsible for the bottleneck, visualizations of θ∇2θ and probability density functions at various Sc values are presented and compared with those of ui∇2ui. The numerical method employed for generating three-dimensional spectra and quantities such as energy and scalar variance dissipation in physical space must be accurate, particularly in resolving small scales. This paper additionally demonstrates that the second-order finite difference scheme conserving kinetic energy and scalar variance in the inviscid limit in viscous simulations accurately predicts the exponential decay range in one-dimensional and three-dimensional turbulent kinetic energy and scalar variance spectra.

A Linear Second-Order Finite Difference Scheme for the Allen–Cahn Equation with a General Mobility

A Linear Second-Order Finite Difference Scheme for the Allen–Cahn Equation with a General Mobility

Finite Difference and Chebyshev Collocation for Time-Fractional and Riesz Space Distributed-Order Advection–Diffusion Equation with Time-Delay

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm.

A compact finite-difference and Haar wavelets collocation technique for parabolic volterra integro-differential equations

Abstract In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations (parabolic-VIDEs) is conducted. The approach focuses on semi-discretizing parabolic-VIDEs by utilizing a second-order compact finite difference method O ( δ t 2 ) for the time variable and approximating the integral term using the trapezoidal rule. Further, the spatial derivative is approximated by Haar wavelet method. This hybrid methodology leverages the strengths of both techniques to solve the equations more efficiently and accurately. The error analysis, using L 2 and L ∞-norms, shows low computational costs. Numerical experiments show that second-order accuracy in both time and space, respectively. Stability and convergence of the proposed method are given through Sobolve space. Some numerical examples are tested for the effectiveness and accuracy of the proposed method. Furthermore, the presented numerical approach is compared with standard finite difference numerical methods and the results are reported in a table.

Identification of Boundary Conditions in a Spherical Heat Conduction Transmission Problem

Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat equations with spherical symmetry. The zero degeneracy at the left interval 0<r<R1 leads to solution difficulties in the one-dimensional interface problem. So, we first investigate the well-posedness of the direct (forward) problem in special weighted Sobolev spaces. Then, we formulate three groups of unknown boundary conditions and inverse problems upon internal point measurements for the heat equation with spherical symmetry. Second-order finite difference scheme approaches for direct and inverse problems are developed. Computational test examples illustrate the theoretical statements proposed.

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Explicit Group Iterative Method with Semi-Approximate Implicit Approach for Solving One-Dimensional Burgers’ Equation

This paper is considering a semi-approximate approach in investigating the Burgers’ problem. The process of the discretization of Burgers’ problem has taken place which it starts with the second-order finite difference in the discretize process together with the linearization part by using the semi-approximate implicit scheme in a way to achieve the approximation equation, thus generating the corresponding linear system equations. Besides, the Gauss-Seidel (GS), Successive Over-Relaxation (SOR) and explicit group (EG) iterative method has combined together with the SOR iterative method, namely as 4-point EGSOR (4EGSOR) has been introduced in this study for resolving the linear system. To assess the proficiency of the suggested methods on the approximation equation, the numerical test has been conducted by considering three parameters, which are computational time, iteration number and maximum absolute error. The findings indicate that the 4EGSOR method outperforms both SOR and GS iterative methods.

An optimal parameterized Newton-type structure-preserving doubling algorithm for impact angle guidance-based 3D pursuer/target interception engagement

An optimal parameterized Newton-type structure-preserving doubling algorithm for impact angle guidance-based 3D pursuer/target interception engagement

Boiling heat transfer by phase-field method

AbstractIn this work, we propose and test the validity of a phase-field method tailored specifically for modeling boiling phenomena. The method relies on numerical solutions of the Navier–Stokes equations coupled with a phase-field method and the energy equation. The continuity and Navier–Stokes equations have been modified introducing a source term that accounts for phase change. Likewise, in the conservative Allen–Cahn equation (phase-field method) a source term that accounts for the volume is introduced. The system of governing equations is solved using a projection-correction method and equations are discretized using a second-order finite difference approach. Thanks to the numerical discretization employed, a constant coefficient Poisson equation for pressure is obtained, which can be efficiently solved using FFT-based direct solvers. The proposed method is validated against several benchmarks: an interface undergoing vaporization at a constant rate, the Stefan problem, the adsorption problem, and the growth of a 2D vapor bubble. For all the benchmarks, the present method well matches with analytical and archival literature results for a wide range of vapor-to-liquid density ratios, from $$\rho _v/\rho _l = 1$$ ρ v / ρ l = 1 down to $$\rho _v/\rho _l \simeq 5 \times 10^{-4}$$ ρ v / ρ l ≃ 5 × 10 - 4 (where $$\rho _v$$ ρ v identifies the vapor density and $$\rho _l$$ ρ l the liquid density).

Construction of Second-Order Finite Difference Schemes for Diffusion-Convection Problems of Multifractional Suspensions in Coastal Marine Systems

Introduction. This paper addresses an initial-boundary value problem for the transport of multifractional suspensions applied to coastal marine systems. This problem describes the processes of transport, deposition of suspension particles, and the transitions between its various fractions. To obtain monotonic finite difference schemes for diffusion-convection problems of suspensions, it is advisable to use schemes that satisfy the maximum principle. When constructing a finite difference scheme that adheres to the maximum principle, it is desirable to achieve second-order spatial accuracy for bothinterior and boundary points of the domain under study. Materials and Methods. This problem presents certain difficulties when considering the boundaries of the geometric domain, where boundary conditions of the second and third kinds are applied. In these cases, to maintain second-order approximation accuracy, an “extended” grid is introduced (a grid supplemented with fictitious nodes). The guidelineis the approximation of the given boundary conditions using the central difference formula, with the exclusion of the concentration function at the fictitious node from the resulting expressions. Results. Second-order accurate finite difference schemes for the diffusion-convection problem of multifractional suspensions in coastal marine systems are constructed. Discussion and Conclusion. The proposed schemes are not absolutely stable, and a detailed analysis of stability and convergence, particularly concerning the grid step ratio, remains an important problem that the author plans to address in the future.

Hybrid multiscale computational approach for heat transfer in GaN semiconductors

Abstract With ongoing advancements in semiconductor technology, understanding thermal transport in semiconductor devices—where phonons are the primary charge carriers—is crucial. This involves a multiscale process in which phenomena at different scales are governed by distinct control equations, necessitating various numerical methods. In this context, this paper introduces a novel multiscale simulation method called the Second-Order Reconstruction Operator Finite Difference Lattice Boltzmann Method (ROsFDLBM). This method effectively combines the Lattice Boltzmann Method (LBM) with the Finite Difference Method (FDM). ROsFDLBM employs reconstruction operators to accurately map the relationship between distribution functions and macroscopic physical quantities, significantly reducing errors caused by the transmission of interface information. The efficacy of this coupling scheme has been confirmed through an interface-coupled thermal diffusion model. Comparative analyses have revealed that the relative error of ROsFDLBM does not exceed 1.2%, demonstrating its superior accuracy. Further simulations of GaN devices have confirmed that traditional macroscopic methods tend to underestimate the temperature in the heat source area at micro-nano scales, with discrepancies reaching up to 60.9 K. Overall, the ROsFDLBM aligns with current research findings and offers an accurate and efficient simulation strategy for addressing complex heat transfer problems.

Fluctuation-driven dynamics of liquid nano-threads with external hydrodynamic perturbations

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise

A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise

Effect of internal fluid and external shear flow on vortex-induced vibration response of drilling riser

Effect of internal fluid and external shear flow on vortex-induced vibration response of drilling riser

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation