Fourth-order Finite Difference Research Articles

A New Very High-Order Finite-Difference Method for Linear Stability Analysis and Bi-orthogonal Decomposition of Hypersonic Boundary Layer Flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003). Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

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 higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

A linearlized mass-conservative fourth-order block-centered finite difference method for the semilinear Sobolev equation with variable coefficients

In this paper, a two-dimensional variable-coefficients semilinear Sobolev equation under Neumann boundary condition is considered, and a novel Crank–Nicolson type linearized fourth-order block-centered finite difference scheme which preserves mass is developed and analyzed. Under local Lipschitz continuous assumption on the nonlinear source term, second-order temporal and fourth-order spatial accuracy are rigorously proved for the primal scalar variable p, its gradient u and its flux q simultaneously. Then, stability under a rough time stepsize condition is proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions and well performance of the proposed scheme.

Thermal effects connected to crystallization dynamics: A lattice Boltzmann study

The possible impact of temperature differences during crystal growth is investigated in this study. The organic molecule considered is mandelic acid, an important component for the pharmaceutical industry. The productivity of generating pure mandelic acid crystals are largely determined by the growth process. Reaction conditions, purity of the components, supersaturation, temperature, but possibly also temperature gradients play a central role during crystal growth. In this study a numerical model based on a hybrid solver combining the lattice Boltzmann method with finite differences is developed to model the crystallization dynamics of (S)-mandelic acid (S-ma) taking quantitatively into account temperature effects. At first, the fourth-order finite-difference method used to model energy and species conservation is validated. Then, comparisons are carried out regarding temperature changes within the single-crystal growth cell. In practice, the molar heat generation at the crystal interface shows only a small effect on the temperature field in the surrounding domain, with temperature differences below 1.5 degree. Finally, the study is extended to investigate the impact of forced convection on the crystal habits while taking into account temperature differences.

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd Boussinesq systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

Numerical Solution of Time-Fractional Schrödinger Equation by Using FDM

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme is utilized for discretizing the spatial term. The unconditional stability and convergence of the scheme in the maximum norm are proved. Finally, we verified the theoretical result with a numerical test.

Analysis of two conservative fourth-order compact finite difference schemes for the Klein-Gordon-Zakharov system in the subsonic limit regime

We propose two conservative fourth-order compact finite difference (CFD4C) schemes and give the rigorous error analysis for the Klein-Gordon-Zakharov system (KGZS) with ε∈(0,1] being a small parameter. In the case 0<ε≪1, i.e., the subsonic limit regime, the solution of KGZS propagates waves with wavelength O(ε) in time and O(1) in space, respectively, with amplitude at O(εα†) with α†=min⁡{α,β+1,2}, where α and β describe the incompatibility between the initial data of the KGZS and the limiting Klein-Gordon equation (KGE) as ε→0+ and satisfy α≥0,β+1≥0. Different from the standard KGZS, the oscillation in time becomes the main difficulty in constructing numerical schemes and giving the error analysis for KGZS in this regime. We prove that the schemes conserve the discrete energies. By defining a discrete ‘energy’, applying cut-off technique and considering the convergence rate of the KGZS to the limiting KGE, we obtain two independent error estimates with the bounds at O(h4/ε1−α⁎+τ2/ε3−α†) and O(h4+τ2+εα⁎) where α⁎=min⁡{1,α,1+β}, τ is time step and h is mesh size. Hence, we get uniformly accurate error estimates with the bounds at O(h4+τ2/(4−α†)) when α≥1 and β≥0, and O(h4α⁎+τ2α⁎/3) when 0<α<1 or −1<β<0. While for α=0 or β=−1, the meshing requirements of CFD4Cs are h=O(ε1/4) and τ=O(ε3/2) for 0<ε≪1. The compact schemes provide much better spatial resolution compared with the usual second order finite difference schemes. Our numerical results support energy-conserving properties and the results of the convergence analysis.

Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options

The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated.

On a sparse and stable solver on graded meshes for solving high-dimensional parabolic pricing PDEs

The purpose of this work is to investigate the multi-dimensional Black-Scholes partial differential equation with variable coefficients numerically. The problem is of practical importance due to option pricing at the presence of multi assets. Since by increasing the dimension, the curse of dimensionality restricts the computations, the proposed solver will be constructed based on sparse arrays. Toward this goal, fourth-order finite difference approximations on graded meshes are introduced and then employed through semi-discretization. Then a sixth-order Runge-Kutta solver is employed for finding the resolution of the derived set of ordinary differential equations. The stability of the proposed scheme is furnished in detail as well. Numerical testings are given to uphold the accuracy and efficacy of the proposed procedure.

Analysis on mathematical model of convection system of micropolar fluid as darcy forchheimer flow undergoes heterogeneous and homogeneous chemical reaction

In this article, to study the fluid behavior on various specific conditions to improve the heat transfer system and here, analysis on mathematical model of dynamic fluid consist of micropolar fluid has allowed micro rotational effect with laminar flow of Darcy forchheimer model which allow inertia effect has incorporated with heterogeneous and homogenous chemical reaction undergoes heat exchanger system with boundary layer problem has modeled. The mathematical model of fluid mechanic governing equations are in the form of partial differential equation (PDE) and similarity transformation into numerical methods (PC4-FDM) of predictor and corrector technique undergoes discretized mesh point and convergence with fourth order finite difference method and shooting method is also equipped to get better solution. The additions of significant heterogeneous parameter depicts that increasing behavior in fluid concentration and homogeneous parameter depicts that decreasing in fluid concentration by allowed micro rotations leads to collision and on increasing the Eckert number related to viscous dissipation has exhibited that increased fluid velocity and decreased fluid temperature. Micro rotation parameter exhibits that similar increased fluid velocity and slight decreased in temperature of the fluid. Darcy forchheimer parameter which related to inertial effect has depicts that decreased in velocity with increased temperature of the fluid in convection system. Due to Industrialization, the study of convection heat transfer system has enormous scope which has necessity to improving the heating and cooling system of industrial mass machineries like powerplant, waste heat recovery unit, pharmaceutical industries etc.

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing

This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The corrections are applied solely to the right-hand side, and leave the finite difference matrix unchanged. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.

Periodic Behaviors of a Linear Fourth-Order Difference Solution to the Benjamin–Bona–Mahony-Type Equation with Time-Periodic Boundaries

The periodic behaviors of a linear fourth-order difference solution to the Benjamin–Bona–Mahony (BBM)-type equation with time-periodic boundaries are analyzed in this paper. Firstly, we employ a variable transformation to change the original BBM-type equation with time-periodic boundaries into a new BBM-type equation with zero boundaries. We then construct a fourth-order linear finite difference method to discrete the new BBM-type equation. The solvability, convergence, stability and accuracy of the approximating solution are discussed. The computation procedure of the present method is given in detail. Numerical results show that the proposed difference method is reliable and efficient for time-periodic simulation.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.