Second-order Finite Difference Method Research Articles

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

A Second-Order Finite-Difference Method for Derivative-Free Optimization

In this paper, a second-order finite-difference method is proposed for finding the second-order stationary point of derivative-free nonconvex unconstrained optimization problems. The forward-difference or the central-difference technique is used to approximate the gradient and Hessian matrix of objective function, respectively. The traditional trust-region framework is used, and we minimize the approximation trust region subproblem to obtain the search direction. The global convergence of the algorithm is given without the fully quadratic assumption. Numerical results show the effectiveness of the algorithm using the forward-difference and central-difference approximations.

A second-order accurate and unconditionally energy stable numerical scheme for nonlinear sine-Gordon equation

In this work, a second-order finite difference method is proposed to solve a nonlinear sine-Gordon equation. The constructed implicit scheme is proved to be unconditionally energy stable. A linear iteration algorithm is used to solve this nonlinear numerical scheme, and we prove that this iteration algorithm is convergent with a negligible constraint for time step. By constructing a suitable high-precision numerical solution, and using the inverse inequality and refinement constraint Δt≤Ch, the error estimate in L∞(0,T;L∞) norm of the fully discrete scheme is obtained. Several numerical examples are given to confirm the sharpness of our theoretical analysis.

A simple model of nutrient recycling and dormancy in a chemostat: Mathematical analysis and a second-order nonstandard finite difference method

A chemostat is an apparatus that sustains a homogeneous environment through continuous inflow and outflow. Presented is a chemostat model that characterizes the dynamics of dormancy-capable microorganisms. This model of coupled systems of nonlinear ordinary differential equations (ODEs) can apply to various types of organisms, such as different species of bacteria, archaea, algae, fungi, viruses, and protozoa. However, these species reside in different environments and rely on different sets of nutrients. Thus, the model adapts to each species’ limiting nutrient through nutrient recycling. This paper includes a complete stability analysis, supporting phase plane portraits, and accompanying bifurcation diagrams. The paper also proposes an advanced second-order, positivity-preserving, and elementary stable nonstandard finite difference method for solving the mathematical model. Series of numerical simulations are presented that support the theoretical results and explore different biological scenarios. The stability analysis reveals that (1) if the overall dilution, death, and conversion are less than the overall growth, both the dormant and active populations persist when introduced to a chemostat; and (2) if the overall dilution, death, and conversion are more than the overall growth, the microorganism population in its entirety will die off when introduced to a chemostat. Furthermore, the model study suggests that neither dormancy nor nutrient recycling provides substantial survival advantages in a basic chemostat when no threat to the active microorganism is present.

Entropy generation in a partially heated hybrid nanofluid saturated wavy porous cavity

PurposeA porous cavity flow field generates entropy owing to energy and momentum exchange within the fluid and at solid barriers. The heat transport and viscosity effects on fluid and solid walls irreversibly generate entropy. This numerical study aims to investigate convective heat transfer together with entropy generation in a partially heated wavy porous cavity filled with a hybrid nanofluid.Design/methodology/approachThe governing equations are nondimensionalized and the domain is transformed into a unit square. A second-order finite difference method is used to have numerical solutions to nondimensional unknowns such as stream function and temperature. This numerical computation is conducted to explore a wide range of regulating parameters, e.g. hybrid nano-particle volume fraction (σ = 0.1%, 0.33%, 0.75%, 1%, 2%), Rayleigh–Darcy number (Ra = 10, 102, 103), dimensionless length of the heat source (ϵ = 0.25, 0.50,1.0) and amplitude of the wave (a = 0.05, 0.10, 0.15) for a number of undulations (N = 1, 3) per unit length.FindingsA thorough analysis is conducted to analyze the effect of multiple factors such as thermal convective forces, heat source, surface corrugation factors, nanofluid volume fraction and other parameters on entropy generation. The flow and temperature fields are studied through streamlines and isotherms. The average Bejan number suggested that entropy generation is entirely dominated by irreversibility due to heat transport at Ra = 10, and the irreversibility due to the viscosity effect is severe at Ra = 103, but the increment in s augments irreversibility due to the viscosity effect over the heat transport at Ra = 102.Originality/valueTo the best of the authors’ knowledge, this numerical study, for the first time, analyzes the influence of surface corrugation on the entropy generation related to the cooling of a partial heat source by the convection of a hybrid nanofluid.

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

Computing the Tracy-Widom Distribution for Arbitrary $\beta>0$

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

Influence of shape errors and inertia effects on the error motion of the aerostatic spindle

In this study, an extended Reynolds equation incorporating the shape errors and the inertia effects of compressible fluid was formulated. The pressure-solving equation for the gas film was constructed using a second-order finite difference method. By combining the measurement data from the cylindricity of multiple shafts obtained from a cylindricity measuring instrument, the variation of the aerostatic bearing load capacity with the rotation of the shaft was calculated, both with and without considering the inertia effect. An experimental rig was also constructed to measure the error motion data of the corresponding shaft. The results of numerical analysis and experimental validation demonstrated that shafts with larger cylindricity error values exhibit greater deviations in the calculated bearing load capacity, which correspondingly leads to larger radial error motion values measured in the experiment. When the roundness error value of the shaft is small or the shaft speed is low, the influence of inertial effects on the calculation results of the load capacity deviation is minimal and can generally be disregarded.

Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application

Water is a weakly compressible fluid medium. Due to its low compressibility, it is usually assumed that water is an incompressible fluid. However, if there are high-pressure pulse waves in water, the compressibility of the water medium needs to be considered. Typical engineering applications include water hammer protection and pulse fracturing, both of which involve the problem of discontinuous pulse waves. Traditional calculation and simulation often use first-order or second-order precision finite difference methods, such as the MacCormark method. However, these methods have serious numerical dissipation or numerical dispersion, which hinders the accurate evaluation of the pulse peak pressure. In view of this, starting from the weakly compressible Navier–Stokes (N-S) equation, this paper establishes the control equations in the form of flux, derives the expressions of eigenvalues, eigenvectors, and flux vectors, and gives a new flux vector splitting (FVS) formula by considering the water equation of state. On this basis, the above flux vector formula is solved using the fifth-order weighted essentially non-oscillatory (WENO) method. Finally, the proposed FVS formula is verified by combining the typical engineering examples of water hammer and pulse fracturing. Compared with the traditional methods, it is proved that the FVS formula proposed in this paper is reliable and robust. As far as we know, the original work in this paper extends the flux vector splitting method commonly used in aerodynamics to hydrodynamics, and the developed model equation and method are expected to play a positive role in the simulation field of water hammer protection, pulse fracturing, and underwater explosion.

Full-rank and low-rank splitting methods for the Swift–Hohenberg equation

The Swift–Hohenberg (SH) equation is an important model in the study of pattern formation. In this paper, we propose two full-rank splitting schemes and a low-rank approximation for the SH equation. We first employ a second-order finite difference method to approximate the space derivatives. Based on the resulting semi-discrete system, two full-rank splitting schemes are derived. The convergences of these schemes are studied. For finding a low-rank solution of the SH equation, we derive a low-rank splitting approximation. Numerical results indicate that our methods are robust, accurate and energy dissipation-preserving.

High-order unconditionally maximum-principle-preserving parametric integrating factor Runge-Kutta schemes for the nonlocal Allen-Cahn equation

We utilize the second-order quadrature-based finite difference method and the high-order parametric integrating factor Runge-Kutta (pIFRK) integrators to construct efficient and accurate schemes for solving the nonlocal Allen-Cahn equation. These schemes preserve the maximum principle for any time-step, and exhibit up to fourth-order accuracy in the temporal direction. We establish a rigorous error estimate and an asymptotic compatibility analysis for the pIFRK schemes. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, verify their asymptotic compatibility, and investigate the discontinuity of the nonlocal Allen-Cahn equation under certain conditions.

Conservative second-order finite difference method for Camassa–Holm equation with periodic boundary condition

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation.

A novel second-order nonstandard finite difference method preserving dynamical properties of a general single-species model

In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments.

Two linearized second-order block-centered finite difference methods for nonlinear Sobolev equations

Two linearized second-order block-centered finite difference methods for nonlinear Sobolev equations

Numerical study of transient convective turbulent boundary layer flow along a vertical plate: analysis of kinetic energy and its dissipation rate

ABSTRACT The present article numerically investigates the turbulent buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model. The deployed turbulence model is appropriate for low Reynolds number (LRN) turbulent flow adjacent to a solid boundary, and the turbulence kinetic energy (TKE) and dissipation rate of TKE are estimated using the momentum equations and are solved simultaneously with the mean flow conservation equations. Two-dimensional time-dependent viscous incompressible turbulent flow is simulated. This flow domain is governed by a highly non-linear group of partial differential equations, namely the time-averaged continuity, momentum, and energy, and also the flow property is determined through TKE, and dissipation rate of TKE equations. Since these equations are not solvable using analytical methods, an implicit second-order finite difference method is employed to solve the governing turbulent flow equations numerically. The simulated time-averaged velocity, temperature, TKE, and dissipation rate of TKE profiles along with friction factor and heat transfer rate are computed for different values of turbulent Reynolds and Prandtl numbers. Average velocity, temperature, turbulence energy, and dissipation rate under both transient and steady-state conditions are decreased with increment in . There is a decrement in average transient velocity and turbulence energy rates with increasing values, whereas the average temperature profile increases. Average steady temperature and turbulence energy profiles are also enhanced with rising values of . The majority of researchers focus on laminar flow and thus far turbulent flow has mainly been addressed through experiments only. To better elucidate the flow in this condition, the novelty of the present study is to utilize a numerical FDM procedure to probe more deeply into the turbulence characteristics in buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model, as this topic is fundamental to many applications in science and engineering. Furthermore, the results of the turbulent flow simulations obtained from the current model are corroborated with existing results for the special case of laminar flow and a good correlation is achieved.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Amplitude behavior in low-rank acoustic modeling

The numerical simulation of wave propagation can be represented by a propagator matrix applied to previous instances of the wavefield. Using the sparsity of the propagator matrix to approximate it by a low-rank representation, one can increment the wavefield’s phase from one time instance to another. Though this procedure does not pay attention to the amplitudes of the seismic waves, it is important to understand its dynamic properties. Here, we evaluate the amplitudes obtained by the low-rank method in the simulation of 2D acoustic wave propagation. In homogeneous media, where theoretical expressions for the wavefield are available, the method provides not only an excellent kinematic approximation, but also reliable amplitudes. For a single horizontal reflector below a homogeneous overburden, the reflection coefficients approximated by the low-rank method are of the same quality or slightly superior to those obtained by a second-order finite-difference (FD) method (implementation from SU). However, in more generally inhomogeneous media, our tests showed larger discrepancies between FD and low-rank modeling results. While comparing unfavorably with FD regarding computation time for small models, its quasi-linear scaling with model size makes the low-rank method superior for large models. Moreover, a generalization to more complex, e.g., anisotropic, media is straightforward.

An efficient numerical scheme and its stability analysis for a time-fractional reaction diffusion model

An efficient numerical scheme is presented to solve a time-fractional reaction–diffusion (TFRD) model with Caputo temporal–fractional derivative. The L1 scheme is employed to approximate the Caputo temporal–fractional derivative and a collocation method based on sextic B-spline (SBS) basis functions is used for discretization of the spatial variable. The stability of the method is investigated. Three numerical examples are provided to illustrate the performance and accuracy of the method. The convergence order of proposed method is O(Δt2−α+Δx6), where α is the order of fractional derivative (FD) (0<α<1) and Δt and Δx represent, respectively, the temporal step size and the spatial mesh size. The influence of the order of fractional derivative α on the solution profile is examined. The computed results and CPU times are compared with those obtained by the standard second-order finite difference method.

A robust computational approach to Lees–Dorodnitsyn laminar hypersonic boundary layers with temperature-dependent properties

The Lees–Dorodnitsyn (L–D) boundary layer equations for two-dimensional, non-reactive, laminar, hypersonic, boundary layer flows, and an assumption of an isentropic external flow are examined. They are applied to various geometries for which the Thin Shear Layer assumptions are valid. This study expands on previous work to develop a novel and robust methodology for computing high-temperature hypersonic flows using a uniform and compact computational stencil implemented through a computational tool, the Bulk-property Boundary Layer (BuBL) solver. In particular, we explore the impact of treating high-temperature effects present in hypersonic flows, namely, treating air as a thermally perfect gas with temperature-variable properties. The ability to solve these flows computationally using second-order finite difference methods is evaluated as are various models for viscosity, Prandtl number, and specific heat. The methodology for solving the external flow properties in the transformed L–D computational domain is also discussed. It is shown that the L–D equations evaluated using the “box” computational stencil are an effective means for evaluating laminar hypersonic boundary layer flows. Solutions for displacement and momentum thicknesses, skin friction, and Stanton number variations are obtained as a function of Prandtl number, specific heat model, and Mach number. Verification and validation measures are performed for the code. Excellent agreement is found in comparisons between BuBL and other computational fluid dynamics and experimental results, thus demonstrating the utility of the proposed methodology.

A second-order finite difference method for the multi-term fourth-order integral–differential equations on graded meshes

A second-order finite difference method for the multi-term fourth-order integral–differential equations on graded meshes