Fourth-order Compact Scheme Research Articles

Hermite Interpolation Approach to High-Order Approximation of Heat Equations


 
 
 It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
 
 

Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers’ equation, from which L1-discretization formula is applied to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete L1 compact difference scheme on uniform meshes can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability of the proposed scheme are rigorously proved in the L∞-norm by the energy argument and mathematical induction. We also establish a temporal second-order compact difference scheme on graded time meshes for solving the problem with weak initial singularity. Finally, several numerical experiments are provided to test the accuracy of two numerical schemes and verify the theoretical analysis.

Finite difference schemes for MHD mixed convective Darcy–forchheimer flow of Non-Newtonian fluid over oscillatory sheet: A computational study

This contribution proposes two third-order numerical schemes for solving time-dependent linear and non-linear partial differential equations (PDEs). For spatial discretization, a compact fourth-order scheme is deliberated. The stability of the proposed scheme is set for scalar partial differential equation, whereas its convergence is specified for a system of parabolic equations. The scheme is applied to linear scalar partial differential equation and non-linear systems of time-dependent partial differential equations. The non-linear system comprises a set of governing equations for the heat and mass transfer of magnetohydrodynamics (MHD) mixed convective Casson nanofluid flow across the oscillatory sheet with the Darcy–Forchheimer model, joule heating, viscous dissipation, and chemical reaction. It is noted that the concentration profile is escalated by mounting the thermophoresis parameter. Also, the proposed scheme converges faster than the existing Crank-Nicolson scheme. The findings that were provided in this study have the potential to serve as a helpful guide for investigations into fluid flow in closed-off industrial settings in the future.

Explicit high-order compact difference method for solving nonlinear hyperbolic equations with three types of boundary conditions

In this paper, an explicit compact difference method is constructed to solve nonlinear hyperbolic equations with initial and three types of boundary conditions. Firstly, a nonlinear explicit fourth-order compact difference scheme is derived by truncation error residual correction and the fourth-order Padé approximation. The corresponding scheme for the linear hyperbolic equation is proven to be conditionally stable by the Fourier analysis method. Then, to improve the computational efficiency, the nonlinear scheme is linearized to obtain a linear explicit fourth-order compact difference scheme. Finally, numerical experiments are conducted to verify the stability and accuracy of the presented method.

A new high-order compact and conservative numerical scheme for the generalized symmetric regularized long wave equations

ABSTRACT In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis.

Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.

Pointwise error analysis of the BDF3 compact finite difference scheme for viscous Burgers' equations

This paper formulates a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme based on a developed fourth-order operator for computing the approximate solution of the Burger's equation. The equation is one of the useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics and etc. This proposed approach approximates the solution of Burger's equation with the help of two main steps. In the first step, the temporal discretization is accomplished by virtue of the BDF3 approach. In the second step, a developed fourth-order operator and the classic compact difference formula combine with the method of order reduction are applied for spatial discretization, thereby constructing a fully-discrete scheme. The theoretical analysis is proved in detail by means of the discrete energy method. The proposed scheme is convergent with third order for time and fourth order for space. Numerical results are carried out to verify the validity and accuracy of the proposed method.

High-order compact finite difference schemes for solving the regularized long-wave equation

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ-weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments.

An efficient extrapolation multigrid method based on a HOC scheme on nonuniform rectilinear grids for solving 3D anisotropic convection–diffusion problems

We develop an efficient multigrid method combined with a high-order compact (HOC) finite difference scheme on nonuniform rectilinear grids for solving 3D diagonal anisotropic convection–diffusion problems with boundary/interior layers. Firstly, we derive a fourth-order compact finite difference scheme to discretize the 3D anisotropic convection–diffusion equation on a rectilinear grid. Then, the resulting large-scale asymmetric linear system of equations is solved by a generalized extrapolation cascadic multigrid (gEXCMG) method based on two novel multigrid (MG) prolongation operators. The highlight of this paper is the application of the quintic Lagrange interpolation and the completed Richardson extrapolation in the design of the new MG prolongation operator on nonuniform rectilinear grids, which can produce a good initial guess (sixth-order approximation to the finite difference solution) for the SSOR-preconditioned biconjugate gradient stabilized (BiCGStab) smoother. In the end, numerical experiments show that the gEXCMG method combined with the HOC scheme can achieve fourth-order accuracy for 3D anisotropic convection–diffusion problems with few smoothing steps on the finest grid. Moreover, the proposed gEXCMG method can offer substantially better efficiency than the state-of-the-art algebraic MG solver, namely, aggregation-based algebraic multigrid (AGMG) method, for large linear systems arising from the discretization of second order elliptic PDEs.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

A second-order energy stable and nonuniform time-stepping scheme for time fractional Burgers' equation

In this paper, an effective finite difference scheme of high order accuracy is proposed for the nonlinear time fractional Burgers' equation. Specifically, we apply the Alikhanov's scheme on graded mesh in the temporal direction and a novel fourth-order compact scheme in the spatial discretization. The proposed scheme resolves initial weak singularity of the solution and preserves high resolution in the space direction. It is rigorously proved that the finite difference scheme is uniquely solvable, discrete variational energy dissipation law and unconditionally stable and convergent in sense of discrete L 2 -norm. With appropriate choice of the grading parameter, the convergence accuracy is min ⁡ { r α , 2 } order in time and fourth order in space, where r is the mesh grading. In the numerical implementation procedure, fast convolution technique and adaptive time-stepping strategy are adopted to accelerate the presented solver and to capture evolution of the solution. Numerical experiments are carried out to verify the validity and effectiveness of the proposed scheme for solving nonlinear time-fractional Burgers' equation.

Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model

American put options with the regime-switching model is a system of coupled free boundary problems. In this study, we present an accurate finite difference method coupled with the Hermite interpolation for solving this system. To this end, we first employ the logarithmic transformation to map the free boundary for each regime to a fixed interval and then eliminate the first-order derivatives in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. We then discretize the system using the fourth-order compact scheme coupled with the Crank–Nicholson method. At the same time, the influence of other asset options and option sensitivities are estimated based on the third-order Hermite interpolation. As such, the overall scheme consists of four tridiagonal linear systems, which can be easily solved using the Thomas algorithm and the Gauss–Seidel iteration. The obtained scheme is then applied for the model with two, four, and sixteen regimes, respectively. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.

Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media

In this article, we present an efficient numerical algorithm, which combines the fourth-order compact difference scheme (CDS) in space with the fast time two-mesh (TT-M) FBN-θ method, to solve the nonlinear distributed-order fractional Sobolev model appearing in porous media. We also construct the corrected CDS by adding the starting part to deal with the problem with nonsmooth solution and recovery the convergence rate. We derive optimal convergence results and prove the stability of the presented numerical algorithm. Finally, we implement numerical experiments by taking several examples with smooth and nonsmooth solutions to verify the correctness of the theoretical results, the effectiveness of the corrected algorithm and the computational efficiency of the fast algorithm.

A fourth-order compact implicit immersed interface method for 2D Poisson interface problems

This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes.

A Third-Order Two-Stage Numerical Scheme for Fractional Stokes Problems: A Comparative Computational Study

Abstract A third-order numerical scheme is proposed for solving fractional partial differential equations (PDEs). The first explicit stage can converge fast, and the second implicit stage is responsible for enlarging the stability region. The fourth-order compact scheme is employed to discretize spatial derivative terms. The stability of the scheme is given for the standard fractional parabolic equation, whereas convergence of the proposed scheme is given for the system of fractional parabolic equations. Mathematical models for heat and mass transfer of Stokes first and second problems using Dufour and Soret effects are given in a set of linear and nonlinear PDEs. Later on, these governing equations are converted into dimensionless PDEs. It is shown that the proposed scheme effectively solves the fractional forms of dimensionless models numerically, and a comparison is also conducted with existing schemes. If readers want it, a computational code for the discrete model system suggested in this paper may be made accessible to them for their convenience.

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with in the discrete -norm is obtained. Here, and are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

An Explicit‐Implicit Numerical Scheme for Time Fractional Boundary Layer Flows

This contribution is concerned with constructing a fractional explicit-implicit numerical scheme for solving time-dependent partial differential equations. The proposed scheme has the advantage over some existing explicit in providing better stability region. But it has one of its limitations of being conditionally stable, even having one implicit stage. For spatial discretization, a fourth-order compact scheme is considered. The stability and convergence of the proposed scheme for respectively the scalar parabolic equation and system of parabolic equations are given. For the sake of application of the scheme, fractional models of flow between parallel plates and mixed convection flow of Stokes' problems under the effects of viscous dissipation and thermal radiation are constructed. The proposed scheme for the classical model is also compared with built-in Matlab solver pdepe for solving parabolic and elliptic equations and existing numerical schemes. It is found that Matlab solver pdepe is failed to find the solution of the considered flow problem with larger values of Eckert number or coefficient of the nonlinear term. But, the proposed scheme successfully finds the solution for classical and fractional models and shows faster convergence than the existing scheme. We provide illustrative computer simulations to show the principal computational features of this approach.

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.

A new linearized fourth-order conservative compact difference scheme for the SRLW equations

A new linearized fourth-order conservative compact difference scheme for the SRLW equations