Richardson Extrapolation Method Research Articles

Numerical Simulation of Asymmetric Merging Flow in a Rectangular Channel

The steady, asymmetric and two-dimensional flow of viscous, incompressible and Newtonian fluid through a rectangular channel with splitter plate parallel to walls is investigated numerically. Earlier, the position of the splitter plate was taken as a centreline of channel but here it is considered its different positions which cause the asymmetric behaviour of the flow field. The geometric parameter that controls the position of splitter is defined as splitter position parameter a. The plane Poiseuille flow is considered far from upstream and downstream of the splitter. This flow-problem is solved numerically by a numerical scheme comprising a fourth order method, followed by a special finite-method. This numerical scheme transforms the governing equations to system of finite-difference equations, which are solved by point S.O.R. iterative method. In addition, the results obtained are further refined and upgraded by Richardson Extrapolation method. The calculations are carried out for the ranges -1 α R 5. The results are compared with existing literature regarding the symmetric case (when a = 0) for velocity, vorticity and skin friction distributions. The comparison is very favourable. Moreover, the notable thing is that the decay of vorticity to its downstream value takes place over an increasingly longer scale of x as R increases for symmetric case but it is not so for asymmetric one.

Benchmark solutions for two-dimensional fluid flow and heat transfer problems in irregular regions using multigrid method

In the previous researches about the benchmark solutions in the irregular regions, there are some obvious shortcomings, such as small number of computational grids, simple computational domain, small characteristic number, and lack of mixed convection benchmark solution. In order to improve this situation, this article deeply studied the benchmark solutions for two-dimensional fluid flow and heat transfer problems in the irregular regions under the body-fitted coordinate system. Taking the lid-driven flow, natural convection, and mixed convection problems as the research objects and considering the influence of boundary configurations of the computational domains, Reynolds number, Prandtl number, and Grashof number, this article designed three groups of test cases. The program code is first verified through employing the nonlinear multigrid method based on the collocated finite volume method and then the numerical solutions of three test cases with 1024 × 1024 grids and convergence criterion of 10−14 are obtained to get the benchmark solutions estimated by the Richardson extrapolation method.

Numerical method for stability analysis of functionally graded beams on elastic foundation

This paper considers the problems of stability of non-uniform and axially functionally graded (FG) Euler–Bernoulli beams on elastic foundation. The boundary conditions of the beam are given in general form covering different classical supporting conditions. Furthermore, the boundary conditions are transformed into convenient form in order to avoid handling infinity values of the stiffness coefficients. The method of initial parameters in differential form is used for the numerical solution of the problem. The solution of the posed boundary value problem is reduced to the iterative solution of two initial value problems and homogeneous linear algebraic system of order two. The Richardson extrapolation method is employed to improve the accuracy of results. The performance of the method proposed has been validated by several numerical examples.

Efficient methods for integrating weight function: a comparative analysis

This paper introduces Romberg-Richardson\'s method as one of the numerical integration tools for computation of stress intensity factor in a pre-cracked specimen subjected to a complex stress field across the crack faces. Also, the computation of stress intensity factor for various stress fields using existing three methods: average stress over interval method, piecewise linear stress method, piecewise quadratic method are modified by using Richardson extrapolation method. The direct integration method is used as reference for constant and linear stress distribution across the crack faces while Gauss-Chebyshev method is used as reference for nonlinear distribution of stress across the crack faces in order to obtain the stress intensity factor. It is found that modified methods (average stress over intervals-Richardson method, piecewise linear stress-Richardson method, piecewise quadratic-Richardson method) yield more accurate results after a few numbers of iterations than those obtained using these methods in their original form. Romberg-Richardson\'s method is proven to be more efficient and accurate than Gauss-Chebyshev method for complex stress field.

On the accuracy of the Haar wavelet discretization method

Current study contains adaption of Haar wavelet discretization method (HWDM) for FG beams and its accuracy estimates. The convergence analysis is performed for differential equations covering a wide class of composite and nanostructures. Corresponding error bound has been derived. It has been shown that the order of convergence of the HWDM can be increased from two to four by applying Richardson extrapolation method. The theoretical estimates are validated by numerical samples considering FGM beam as a model problem. The results obtained by applying HWDM are compared with the results of finite difference method (FDM).

A Local Fractional Taylor Expansion and Its Computation for Insufficiently Smooth Functions

AbstractA general fractional Taylor formula and its computation for insufficiently smooth functions are discussed. The Aitken delta square method and epsilon algorithm are implemented to compute the critical orders of the local fractional derivatives, from which more critical orders are recovered by analysing the regular pattern of the fractional Taylor formula. The Richardson extrapolation method is used to calculate the local fractional derivatives with critical orders. Numerical examples are provided to verify the theoretical analysis and the effectiveness of our approach.

A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of Open image in new window, where N + 1 and N0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.

Convergence theorem for the Haar wavelet based discretization method

The accuracy issues of Haar wavelet method are studied. The order of convergence as well as error bound of the Haar wavelet method is derived for general nth order ODE. The Richardson extrapolation method is utilized for improving the accuracy of the solution. A number of model problems are examined. The numerically estimated order of convergence has been found in agreement with convergence theorem results in the case of all model problems considered.

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.

A two-grid method for an elliptic equation with boundary layers on a Shishkin mesh

A linear elliptic equation with regular and parabolic boundary layers is considered. It is solved by using an upwind difference scheme on a Shishkin mesh with the property of uniform convergence with respect to a small parameter. The problem of reducing the number of arithmetic operations required to implement the difference scheme on the basis of a two-grid method is investigated. The application of the Richardson extrapolation method to increase the accuracy of the difference scheme is studied. Results of numerical experiments are presented.

Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method (IIM) is proposed to solve two-dimensional (2D) heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact (HOC) difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit (ADI) scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.

Numerical simulation on impingement and film composite cooling of blade leading edge model for gas turbine

In this paper numerical simulation is performed to simulate the impingement and film composite cooling on blade leading edge region. The relative performances of turbulence models are compared with available experimental data, and the results show that SST k–ω model is the best one based on simulation accuracy. Then the SST k–ω model is adopted for the simulation. The grid independence study is also carried out by using the Richardson extrapolation method. A single array of circle jets and three rows of film holes are arranged to investigate the impingement and film composite cooling. Five different blowing ratios and five different film hole spanwise angles are studied in detail. The results indicate that the heat transfer coefficient on the internal surface of turbine blade leading edge increases with the blowing ratio, and slightly changes with film hole spanwise angle. And the external film cooling effectiveness distribution would vary rapidly with the blowing ratio and the film hole spanwise angle.

Estimating uncertainties in statistics computed from direct numerical simulation

Rigorous assessment of uncertainty is crucial to the utility of direct numerical simulation (DNS) results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier–Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow at Reτ ≈ 180. Both small (Lx/δ × Lz/δ = 4π × 2π) and large (Lx/δ × Lz/δ = 12π × 4π) domain sizes are investigated. For each case, a sequence of meshes was generated by first designing a “nominal” mesh using standard heuristics for wall-bounded simulations. These nominal meshes were then coarsened to generate a sequence of grid resolutions appropriate for the Bayesian Richardson extrapolation method. In addition, the small box case is computationally inexpensive enough to allow simulation on a finer mesh, enabling the results of the extrapolation to be validated in a weak sense. For both cases, it is found that while the sampling uncertainty is large enough to make the order of accuracy difficult to determine, the estimated discretization errors are quite small. This indicates that the commonly used heuristics provide adequate resolution for this class of problems. However, it is also found that, for some quantities, the discretization error is not small relative to sampling error, indicating that the conventional wisdom that sampling error dominates discretization error for this class of simulations needs to be reevaluated.

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L2- and H1-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.

Computational fluid dynamics (CFD) mesh independency techniques for a straight blade vertical axis wind turbine

This paper numerically investigates four methods, namely mesh refinement, General Richardson Extrapolation (GRE), Grid Convergence Index (GCI), and the fitting method, in order to obtain a mesh independent solution for a straight blade vertical axis wind turbine (SB-VAWT) power curve using computational fluid dynamics (CFD). The solution is produced by employing the 2D Unsteady Navier–Stokes equations (URANS) with two turbulence models (Shear Stress Transport (SST) Transitional and ReNormalized Groups (RNG) κ−ɛ models). The commonly applied mesh refinement is found to be computationally expensive and not often practical even for a full 2D model of the turbine. The mesh independent power coefficient produced using the General Richardson Extrapolation method is found to be encouraging. However, the Grid Convergence Index may not be applicable in mesh independency tests due to the oscillatory behaviour of the convergence for the turbine power coefficient. As an alternative, the fitting method shows a good potential for the predicting of the mesh independent power coefficient without the necessity to consider a massive number of meshes.

Grid convergence errors in hemodynamic solution of patient-specific cerebral aneurysms

Computational fluid dynamics (CFD) has become a cutting-edge tool for investigating hemodynamic dysfunctions in the body. It has the potential to help physicians quantify in more detail the phenomena difficult to capture with in vivo imaging techniques. CFD simulations in anatomically realistic geometries pose challenges in generating accurate solutions due to the grid distortion that may occur when the grid is aligned with complex geometries. In addition, results obtained with computational methods should be trusted only after the solution has been verified on multiple high-quality grids. The objective of this study was to present a comprehensive solution verification of the intra-aneurysmal flow results obtained on different morphologies of patient-specific cerebral aneurysms. We chose five patient-specific brain aneurysm models with different dome morphologies and estimated the grid convergence errors for each model. The grid convergence errors were estimated with respect to an extrapolated solution based on the Richardson extrapolation method, which accounts for the degree of grid refinement. For four of the five models, calculated velocity, pressure, and wall shear stress values at six different spatial locations converged monotonically, with maximum uncertainty magnitudes ranging from 12% to 16% on the finest grids. Due to the geometric complexity of the fifth model, the grid convergence errors showed oscillatory behavior; therefore, each patient-specific model required its own grid convergence study to establish the accuracy of the analysis.

A comparative study of stress integration methods for the Barcelona Basic Model

The paper compares the accuracy and stability of implicit and explicit stress integration schemes applied to the Barcelona Basic Model. In addition, the effect of the integration scheme on the convergence of Newton–Raphson algorithm is studied. By running a Newton–Raphson algorithm for a single stress point, the number of iterations required to reach convergence gives some insight on the convergence of the finite element solution at the global level. The explicit algorithms tested incorporate substepping with error control and are based on Runge–Kutta methods of different orders or on the Richardson extrapolation method. The implicit return-mapping algorithms follow the procedures of Simo and Hughes [1] and Hofmann [2].

Using Richardson extrapolation techniques to price American options with alternative stochastic processes

In this paper the authors investigate the performance of the original and repeated Richardson extrapolation methods for American option pricing by implementing both the original and modified Geske–Johnson approximation formulae. A comprehensive numerical comparison includes alternative stochastic processes of the underlying asset price. The numerical results show that whether the original or modified formula is implemented, the Richardson extrapolation techniques work very well. The repeated Richardson extrapolation strongly outperforms the original, especially when the underlying asset price follows a stochastic volatility process. Moreover, this study verifies the feasibility of the estimated error bounds of the American option prices under alternative stochastic processes by applying the repeated Richardson extrapolation method and estimating the interval of true American option values, as well as determining the number of options needed for an approximation to achieve a desired accuracy level.

Extrapolation of Mixed Finite Element Approximations for the Maxwell Eigenvalue Problem

In this paper, a general method to derive asymptotic error expansion formulas for the mixed finite element approximations of the Maxwell eigenvalue problem is established. Abstract lemmas for the error of the eigenvalue approximations are obtained. Based on the asymptotic error expansion formulas, the Richardson extrapolation method is employed to improve the accuracy of the approximations for the eigenvalues of the Maxwell system from to when applying the lowest order Nedelec mixed finite element and a nonconforming mixed finite element. To our best knowledge, this is the first superconvergence result of the Maxwell eigenvalue problem by the extrapolation of the mixed finite element approximation. Numerical experiments are provided to demonstrate the theoretical results.

Extrapolation of Mixed Finite Element Approximations for the Maxwell Eigenvalue Problem

In this paper, a general method to derive asymptotic error expansion formulas for the mixed finite element approximations of the Maxwell eigenvalue problem is established. Abstract lemmas for the error of the eigenvalue approximations are obtained. Based on the asymptotic error expansion formulas, the Richardson extrapolation method is employed to improve the accuracy of the approximations for the eigenvalues of the Maxwell system from to when applying the lowest order Nedelec mixed finite element and a nonconforming mixed finite element. To our best knowledge, this is the first superconvergence result of the Maxwell eigenvalue problem by the extrapolation of the mixed finite element approximation. Numerical experiments are provided to demonstrate the theoretical results.