Compact Finite Difference Scheme Research Articles

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

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative methods that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes.

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

Numerical Investigation of Hybrid Nanofluid Natural Convection and Entropy Generation in a Corrugated Enclosure with an Inner Conducting Block

Current work deals with a numerical analysis of convective heat transfer and entropy generation inside a rectangular cavity with a corrugated bottom filled with MoS2–SiO2-water hybrid nanofluid. Here, a conducting solid body is attached to the top wall, and discrete heaters are attached to the bottom wall of the cavity. The numerical solutions of the governing equations are derived utilizing a higher-order compact (HOC) finite difference scheme and validated with the existing computational and experimental results. Present numerical results are then studied in detail, emphasizing isotherms, streamlines, and local entropy generation with respect to specific parameters like Rayleigh number (103 ≤ Ra ≤ 106), the volume percentage of nanoparticles (0% ≤ Φ ≤ 4%), the thermal conductivity of solid body (1.95 ≤ ks ≤ 16.00) as well as the aspect ratio of heater length (AR = 0.2, 0.4, 0.6, 0.8). The impacts of key factors on the Bejan number, average Nusselt number, and overall entropy generation are also investigated. The results show that an increase in the thermal conductivity of the solid body from 1.95 to 16.00 increases the average Nusselt number and total entropy generation by 9.17% and 40.07%, respectively, for AR = 0.2, Ra = 106, and Φ = 4%. In addition, the average Nusselt number and total entropy generation decrease by 59.11% and 61.99%, respectively, for ks = 16.00, Ra = 106, and Φ = 4% when the aspect ratio of heater length increases to 0.8.

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

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.

Compact finite difference schemes with high resolution characteristics and their applications to solve Burgers equation

Compact finite difference schemes with high resolution characteristics and their applications to solve Burgers equation

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

A fast compact finite difference scheme for the fourth-order diffusion-wave equation

In this paper, the H 2 N 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When K 1 < 3 2 , the convergence order is O ( τ 3 − γ + h 4 + ϵ ) , where K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method

&lt;abstract&gt;&lt;p&gt;The 1D and 2D spatial compact finite difference schemes (CFDSs) for time-fractional diffusion equations (TFDEs) were presented in this article with uniform temporal convergence order. Based on the idea of the modified block-by-block method, the CFDSs with uniform temporal convergence order for TFDEs were given by combining the fourth-order CFDSs in space and the high order scheme in time. The stability analysis and convergence order of CFDSs with uniform convergence order in time for TFDEs strictly proved that the provided uniform accuracy time scheme is $ (3-\alpha) $ temporal order and spatial fourth-order, respectively. Ultimately, the astringency of 1D and 2D spatial CFDSs was verified by some numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Direct numerical simulation of turbulence amplification in a strong shock-wave/turbulent boundary layer interaction

In this study, the turbulence amplification mechanism within the strong shock-wave/turbulent boundary layer interaction is investigated using direct numerical simulation (DNS) over a 24° compression ramp with Mach 2.9 flow. A new in-house solver based on the compact finite difference scheme is introduced, and its accuracy is validated by comparing the flow statistics with existing DNS and experimental data. Within the DNS findings, two distinct turbulence kinetic energy (TKE) hotspots are identified. In contrast to previous studies, this study sheds light on shocklets, characterized by mid-frequency features, as a key factor contributing to the second TKE amplification, which occurs near the reattachment point. Streamline coordinate analysis reveals that shear effects dominate TKE production over the flow deceleration effect in the shock-wave/turbulent boundary layer interaction. The shear effect induced by the rolling up of the boundary layer initiates the first TKE amplification near the wall region in proximity to the separation point, followed by flow deceleration due to the main shock wave contributing to TKE generation. The initial detachment of the shear layer enhances the shear contribution. While TKE decreases above the separation bubble due to the positive mean velocity gradient, TKE amplifies again due to the flow deceleration caused by the secondary shock wave. In addition, the intermittently spawning shocklets above the bulge structures enhance the shear effect on the TKE production. Moreover, the generated TKE subsequently transfers to the local pressure minimum line, created by the bulges effect, thereby establishing a spatially converged maximum TKE line.

Compact Finite Difference Scheme for Euler–Bernoulli Beam Equation with a Simply Supported Boundary Conditions

Compact Finite Difference Scheme for Euler–Bernoulli Beam Equation with a Simply Supported Boundary Conditions

Numerical analysis of an uncoupled and linearized compact difference scheme with energy dissipation property for the generalized dissipative symmetric regularized long‐wave equations

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‐norm for both and . Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Analysis of the Dominant Structures of an Impinging Round Jet at M=0.9

Large-eddy simulation of subsonic round jet impinging on a flat circular plate is performed at Mach number 0.9 and Reynolds number 25,000, based on the diameter and centerline velocity of the jet for two impingement distances h of 6r0 and 8r0 using high-order compact finite difference scheme. The complex flow phenomena associated with the impinging jet flowfield are studied. The power spectral density (PSD) of instantaneous pressure reveals a discrete impingement tone. Dynamic mode decomposition (DMD) of the velocity and pressure is also done for both cases to extract the most dominant modes of the flow. The fundamental impingement tone frequency obtained from the PSD of pressure is identical to that of the first dominant pressure mode obtained from DMD. Both axisymmetric and semihelical modes are present for the h=8r0 case, but only axisymmetric modes are there in the h=6r0 case, and it is noted that the modes corresponding to the fundamental tone have an axisymmetric structure. Also, one dominant mode showing the wall jet structures denoting the liftoff of fluid from the wall after impingement at some radial distance away from the stagnation region is present in both impingement distance cases.

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

Design and analysis of efficient computational techniques for solving a temporal‐fractional partial differential equation with the weakly singular solution

This work deals with the construction of robust numerical schemes for solving a time‐fractional convection‐diffusion (TFCD) equation with variable coefficients subject to weakly singular solution. The solution of the underlying problem exhibits a weak singularity at time . The methods are designed on both graded and adapted grids, in which the temporal derivative is approximated on the nonuniform grid, in order to tackle the singularity. In the case of adaptive mesh technique, the mesh is constructed adaptively via equidistribution of a positive monitor function. The resulting semi‐discretized equation is further discretized by a high‐order compact finite difference (CFD) scheme to obtain a fully discrete scheme. Stability and convergence of the method on graded mesh are analyzed. Numerical results for one test problem subjected to nonsmooth analytical solution are presented which shows the robustness and efficiency of the proposed numerical schemes. The elapsed computational time (CPU time) for the proposed methods are provided. Numerical results obtained from the methods on graded and adapted grids are compared with those from the scheme on uniform grid. Finally, as an application, the TFCD model and the suggested numerical techniques were employed to price options of a European butterfly spread. The effect of fractional order (FO) derivative on the option price profile is investigated.

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.