Fourth-order Compact Scheme Research Articles

Solution of high order compact discretized 3D elliptic partial differential equations by an accelerated multigrid method

A robust and efficient numerical solver for three-dimensional linear elliptic partial differential equations without cross derivative terms, but variable coefficients is presented. The proposed algorithm is based on a general meshsize fourth-order compact finite-difference scheme and an accelerated geometric multigrid solver. The former not only increases the accuracy of the discretization, but also does not affect the bandwidth of the common second-order finite-difference scheme, which is relevant for distributed memory parallel implementations of the multigrid preconditioner. The acceleration technique, on the other hand, allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence. Furthermore, it is found that depending on the coefficients of the elliptic partial differential equation, the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence.

Large Eddy Simulation of turbulent natural convection in an inclined tall cavity

To a better understanding of buoyancy-driven flows in enclosed tall cavity, Large Eddy Simulations are performed with an original solver, which combines fourth-order compact scheme and parallel computing. The direct numerical simulation of the natural convection flow between two heated parallel plates are provided to validate the present numerical method. Then, the mean and turbulent statistics of the buoyancy-driven in a tall cavity at the moderate Rayleigh number are computed. The results of the vertical configuration flow are successfully compared with the available experimental data. Then, the statistics are discussed with respect to the inclination angle of the tall cavity. Especially, the occurrence of convection rolls in the cavity inclined to the horizontal is shown, as well as their roles in the deterioration of the thermal stratification.

An efficient, high-order method for solving Poisson equation for immersed boundaries: Combination of compact difference and multiscale multigrid methods

A new efficient and high-order accurate sharp-interface method for solving the Poisson equation on irregular domains and non-uniform meshes is presented. The approach is based on a combination of a fourth-order compact finite difference scheme and a multiscale multigrid (MSMG) method. The key aspect of the new method is that the regular compact finite difference stencil is modified at the irregular grid points near the immersed boundary to obtain a sharp interface solution while maintaining the formal fourth-order accuracy. The MSMG method is designed based on the standard multigrid V-cycle technique to solve the system of equations derived from the fourth-order compact discretization, while the corresponding multigrid relaxation, restriction and prolongation operators are properly constructed for non-uniform grids with immersed boundaries. The contribution of the present work is the design of a fourth-order-accurate Poisson solver whose accuracy, efficiency and computational cost are independent of the complexity of the geometry and the presence or not of an immersed boundary. The new method is demonstrated and validated for a number of problems including smooth and jagged boundaries. The test cases confirm that the new method is fourth-order accurate in the maximum norm whether an immersed boundary is present or not and on uniform or non-uniform meshes. Furthermore, the computational efficiency of the new method is demonstrated with regard to convergence rate and run time, which shows that the MSMG method is equally efficient for domains with immersed boundaries as for simple domains. The new compact difference method is evaluated by comparison with the standard fourth-order (non-compact) finite difference approximation in terms of both accuracy and computational efficiency. The new compact difference scheme yields indeed more accurate numerical solutions. The striking difference between the two schemes is the much higher computational efficiency: The number of V-cycles needed to reach the discretization error is significantly lower for the new compact method compared to the standard difference scheme. As a result, the new compact scheme requires only a fraction of the computer time for convergence in comparison to the standard fourth-order difference scheme.

Unconditional $$L^{\infty }$$ L ∞ -convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

This paper is concerned with the unconditional and optimal $$L^{\infty }$$ -error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrodinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of $$O(h^4+\tau ^2)$$ with time step $$\tau $$ and mesh size h in the discrete $$L^{\infty }$$ -norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrodinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.

Maximum norm error estimates of fourth-order compact difference scheme for the nonlinear Schr\xf6dinger equation involving a quintic term

A compact finite difference (CFD) scheme is presented for the nonlinear Schrödinger equation involving a quintic term. The two discrete conservative laws are obtained. The unconditional stability and convergence in maximum norm with order O({tau }^{2}+h^{4}) are proved by using the energy method. A numerical experiment is presented to support our theoretical results.

Fourth-order compact finite difference method for solving two-dimensional convection\u2013diffusion equation

A fourth-order compact finite difference scheme of the two-dimensional convection–diffusion equation is proposed to solve groundwater pollution problems. A suitable scheme is constructed to simulate the law of movement of pollutants in the medium, which is spatially fourth-order accurate and temporally second-order accurate. The matrix form and solving methods for the linear system of equations are discussed. The theoretical analysis of unconditionally stable character of the scheme is verified by the Fourier amplification factor method. Numerical experiments are given to demonstrate the efficiency and accuracy of the scheme proposed, and these show excellent agreement with the exact solution.

Development of a high-order compact finite-difference total Lagrangian method for nonlinear structural dynamic analysis

• A compact finite-difference total Lagrangian method (CFDTLM) is developed for nonlinear structural dynamics analysis. • The large deformation of elastic structures for Hookean and neo-Hookean materials is accurately simulated using the CFDTLM. • The CFDTLM is extended and assessed for simulating thermo-elastodynamic problems. • The numerical solution procedure adopted is simpler than high-order finite volume and finite element formulations . • The solution methodology proposed is accurate for simulating nonlinear structural dynamics problems. A high-order compact finite-difference total Lagrangian method (CFDTLM) is developed and applied to nonlinear structural dynamic analysis. The two-dimensional simulation of thermo-elastodynamics is numerically performed in generalized curvilinear coordinates by taking into account the geometric and material nonlinearities. The spatial discretization is carried out by a fourth-order compact finite-difference scheme and an implicit second-order accurate dual time-stepping method is applied for the time integration. The accuracy and capability of the proposed solution methodology for the nonlinear structural analysis is investigated through simulating different static and dynamic benchmark problems including large deformations, large displacements and Hookean/neo-Hookean materials. The solution method is demonstrated to be free of shear-locking behavior. The results obtained by the present solution algorithm are compared with the analytical solution and the numerical results of the finite element and finite volume methods to examine the accuracy and robustness of the solution method proposed. A grid study is also performed to investigate the grid size effect on the accuracy and performance of the solution. Indications are that the solution methodology proposed is accurate for simulating nonlinear structural dynamics problems.

A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier–Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate flow variables at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the cell averaged slopes inside each control volume through the differences of flow variables at the cell interfaces can be updated as well in GKS. So, with the updated flow variables and their slopes inside each cell, the HWENO techniques can be naturally implemented for the compact high-order reconstruction at the beginning of each time step. Therefore, a compact higher-order GKS, such as the two-stage fourth-order compact scheme can be constructed. This fourth-order compact GKS has the same stencil and as robust as the second-order scheme. In comparison with the fourth-order DG method, they have the same stencil. In order to get a fourth-order temporal accuracy, the GKS uses two stages and DG needs four stages in the time stepping methods. The CFL number used in GKS is on the order of 0.5 instead of 0.11 in the DG. This research concludes that beyond the first-order Riemann solver the use of high-order time evolution model at a cell interface is extremely helpful in the design of robust, accurate, and efficient higher-order compact schemes for the compressible flow simulations.

Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problem

ABSTRACTThis work introduces a new numerical solution to the inverse parabolic problem with source control parameter that has important applications in large fields of applied science. We expand the approximate solution of the inverse problem in terms of shifted Chebyshev polynomials in time and radial basis functions with symmetric variable shape parameter in space, with unknown coefficients. Unknown coefficient matrix determined using the collocation technique. Sample results show that the proposed method is very accurate. Moreover, the proposed method is compared with two other methods, fourth-order compact difference scheme and method of lines. Finally, we examine the stability of our method for the case where there is additive noise in input data.

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection\u2013diffusion equations

In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V- or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.

Application of High-Order Compact Difference Scheme in the Computation of Incompressible Wall-Bounded Turbulent Flows

In the present work, a highly efficient incompressible flow solver with a semi-implicit time advancement on a fully staggered grid using a high-order compact difference scheme is developed firstly in the framework of approximate factorization. The fourth-order compact difference scheme is adopted for approximations of derivatives and interpolations in the incompressible Navier–Stokes equations. The pressure Poisson equation is efficiently solved by the fast Fourier transform (FFT). The framework of approximate factorization significantly simplifies the implementation of the semi-implicit time advancing with a high-order compact scheme. Benchmark tests demonstrate the high accuracy of the proposed numerical method. Secondly, by applying the proposed numerical method, we compute turbulent channel flows at low and moderate Reynolds numbers by direct numerical simulation (DNS) and large eddy simulation (LES). It is found that the predictions of turbulence statistics and especially energy spectra can be obviously improved by adopting the high-order scheme rather than the traditional second-order central difference scheme.

High order method for Black–Scholes PDE

In this paper, the Black–Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than O(h4+δ4). The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets.

Compact finite difference schemes for the time fractional diffusion equation with nonlocal boundary conditions

Fourth-order compact finite difference schemes for solving the time fractional diffusion equation with nonlocal boundary conditions are considered in this paper. The second-order derivative in one space variable is approximated by a fourth-order compact finite difference. Two different time derivatives, the Caputo derivative and Riemann–Liouville derivative are considered in the equations that we discuss. The Caputo time derivative is discretized by the L1-approximation and a second-order approximation is used for the Riemann–Liouville derivative. Stability analysis using the discrete energy method for Scheme I is given, and error estimation of this scheme is derived by conducting the local truncation error analysis, using the stability result. Numerical results are provided to show that both two fully discrete schemes have fourth-order accuracy in space, and $$2-\gamma $$ order of accuracy in time for Scheme I with $$\gamma $$ being the order of the fractional derivative, while Scheme II has second-order accuracy in time when the theoretical solutions are smooth with zero initial values.

Assessment of Characteristic Boundary Conditions Based on the Artificial Compressibility Method in Generalized Curvilinear Coordinates for Solution of the Euler Equations

Abstract The characteristic boundary conditions are applied and assessed for the solution of incompressible inviscid flows. The two-dimensional incompressible Euler equations based on the artificial compressibility method are considered and then the characteristic boundary conditions are formulated in the generalized curvilinear coordinates and implemented on both the far-field and wall boundaries. A fourth-order compact finite-difference scheme is used to discretize the resulting system of equations. The solution methodology adopted is more suitable for this assessment because the Euler equations and the high-accurate numerical scheme applied are quite sensitive to the treatment of boundary conditions. Two benchmark test cases are computed to investigate the accuracy and performance of the characteristic boundary conditions implemented compared to the simplified boundary conditions. The sensitivity of the solution obtained by applying the characteristic boundary conditions to the different numerical parameters is also studied. Indications are that the characteristic boundary conditions applied improve the accuracy and the convergence rate of the solution compared to the simplified boundary conditions.

A Fourth-Order Compact Finite Difference Scheme to the Numerical Solution of FitzHugh-Nagumo Equation

The FitzHugh-Nagumo equation is an important nonlinear reaction-diffusion equation used in physics and chemicals. To obtain the numerical solution of partial differential equations, the compact finite difference method is widely applied. In this paper, I propose a new numerical solution to FitzHugh-Nagumo equation by using a fourth-order compact finite difference scheme in space, and a semi-implicit Crank-Nicholson method in time. I further calculate the results in terms of accuracy by leveraging the proposed method and exact solution. In particular, I compare the new method whose convergence order is close to four with the second order central difference method. The simulated results show the new solution is more accurate and effective. The proposed method is expected to be a good solution to some problems in the real world.

On Coarse Grids Simulation of Compressible Mixing Layer Flows Using Vorticity Confinement

In this work, the capability and performance of the vorticity confinement (VC) implemented in a high-order accurate flow solver in predicting two-dimensional (2D) compressible mixing layer flows on coarse grids are investigated. Here, the system of governing equations with incorporation of the VC in the formulation is numerically solved by the fourth-order compact finite difference scheme. To stabilize the numerical solution, a low-pass high-order filter is applied, and the nonreflective boundary conditions are used at the farfield and outflow boundaries to minimize the reflections. At first, the numerical results without applying the VC are validated by available direct numerical simulations (DNSs) for a low Reynolds number mixing layer. Then, the calculations using a range of VC levels are performed for a high Reynolds number mixing layer and the results are thoroughly compared with those of available large eddy simulations (LESs). The study shows that, with applying the vortex identification method, more accurate results are obtained in the slow laminar region of the mixing layer. A sensitivity study is also performed to examine the effect of different numerical parameters to reasonably provide more accurate results. It is shown that the local VC introduced based on the artificial viscosity coefficient and the vorticity thickness can improve the accuracy of the results in the turbulent region of the mixing layer compared with those of LESs. It is found that the solution methodology proposed can reasonably preserve the vortices in the flowfield and the results are comparable with those of LESs on fairly coarser grids and thus the computational costs can be considerably decreased.

Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

The Conservative Splitting High-Order Compact Finite Difference Scheme for Two-Dimensional Schrödinger Equations

In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrödinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the scheme satisfies the charge and energy conservations and it is unconditionally stable. We also prove the optimal error estimate of fourth-order accuracy in spatial step and second-order accuracy in time step. The scheme can be easily implemented and extended to higher dimensional problems. Numerical examples are presented to confirm our theoretical results.

An efficient parallel high-order compact scheme for the 3D incompressible Navier–Stokes equations

ABSTRACTThis article provides a strategy for solving incompressible turbulent flows, which combines compact finite difference schemes and parallel computing. The numerical features of this solver are the semi-implicit time advancement, the staggered arrangement of the variables and the fourth-order compact scheme discretisation. This is the usual way for solving accurately turbulent incompressible flows. We propose a new strategy for solving the Helmholtz/Poisson equations based on a parallel 2d-pencil decomposition of the diagonalisation method. The compact scheme derivatives are computed with the parallel diagonal dominant (PDD) algorithm, which achieves good parallel performances by introducing a bounded numerical error. We provide a new analysis of its effect on the numerical accuracy and conservation features. Several numerical experiments, including two simulations of turbulent flows, demonstrate that the PDD algorithm maintains the accuracy and conservation features, while conserving a good parallel performance, up to 4096 cores.

Fourth-Order Compact Finite Difference Scheme for American Option Pricing Under Regime-Switching Jump-Diffusion Models

In this paper, a fourth-order compact finite difference scheme is proposed for the solution of partial integro-differential equation arising in option pricing under jump-diffusion models. In proposed compact scheme, second derivative approximations of the unknowns are eliminated with the unknowns itself and their first derivative approximations while retaining the fourth order accuracy and tri-diagonal nature of the scheme. We use the proposed compact scheme with three time levels along with operator splitting technique to solve the linear complementary problem which arises in American option pricing under regime-switching jump-diffusion models. Moreover, it is shown that the proposed scheme leads to a tri-diagonal system of linear equations and fourth order accuracy is obtained. Since initial condition is not smooth enough for jump-diffusion model, we employ the smoothing operators to ensure high-order convergence rate. Numerical examples for American option pricing under Merton and Kou regime-switching jump-diffusion models are given.