Richardson Extrapolation Method Research Articles

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves

In the present study, the added resistance, heave, and pitch of the KRISO Container Ship (KCS) in waves, at both model scale and full scale, are predicted numerically in regular head waves, for four wavelengths and three wave heights. The ISIS-CFD viscous flow solver, implemented in the Fidelity Fine Marine software provided by CADENCE, was employed for the numerical simulations. The spatial discretization was based on the finite volume method using an unstructured grid. The unsteady Reynolds-averaged Navier–Stokes (RANS) equations were solved numerically, with the turbulence modeled by shear stress transport (k-ω) (SST). The free-surface capturing was based on the volume-of-fluid method. The computed solutions were validated through comparisons with towing test data available in the public domain. To predict the uncertainties in the numerical solution, a systematic grid convergence study based on the Richardson extrapolation method was performed for a single wave case on three different grid resolutions. Specific attention was given to the free-surface and wake flow in the propeller plane. The purpose was to compare the numerical results from the model- and full-scale tests to examine the scale’s effect on the ship’s performance in regular head waves. The comparison between the model scale and full scale showed obvious differences, less accentuated for the free-surface topology and clearly observed in terms of boundary layer formation in the propeller’s vicinity.

Verification of the γ-Reθt Transition Model in OVERFLOW and FUN3D

The findings of the recent AIAA and NATO workshops on transition modeling have underscored the importance of verifying the transition models that are based on Reynolds-averaged Navier–Stokes. To that end, the present work aims to verify the Langtry–Menter (LM2009) transition model coupled with Menter’s shear-stress transport (SST) turbulence model. Specifically, the SST-2003-LM2009 model, as implemented in the OVERFLOW structured overset solver and the FUN3D unstructured solver, is applied to both natural and bypass transition scenarios on a flat plate and to natural and separation-bubble-induced transition on the NLF-0416 airfoil. The Richardson extrapolation method was used to analyze and report the grid convergence of integrated and local load coefficients upstream, within, and downstream of the transition region, based on solutions obtained using the two solvers on identical grids. Reasonable agreement is obtained between the two codes for both global and local surface load coefficients when further refining the workshop mesh series, although with a grid-convergence index for the local friction coefficient that may exceed 1% in certain cases. The study also highlights the solution sensitivity to the choice of turbulence inflow boundary conditions and model options, along with the necessity to unambiguously specify this information for benchmark exercises.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

Richardson extrapolation method for solving the Riesz space fractional diffusion problem

AbstractThe Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.

Numerical comparison and efficiency analysis of three vertical axis turbine of H-Darrieus type

Hydropower is an important source of energy in Latin America. Many countries in the region, including Brazil, Peru, Colombia, and Chile, rely heavily on hydropower plants to meet their energy needs. However, there are also challenges related to the use of hydropower in the region, such as the construction of dams that can have negative impacts on ecosystems and local communities. A new alternative is the production of energy through hydrokinetic turbines because they are a clean and renewable energy source that does not emit greenhouse gases. In addition, its production is predictable and can be generated in a variety of environments, from coasts to rivers and canals. Within the hydrokinetic turbines are the H-Darrieus turbines although they are still under development, they are seen as an important opportunity to diversify the energy matrix and reduce dependence on fossil fuels. The main purpose of this study is to determine and compare the efficiency of three Darrieus H-type vertical axis hydrokinetic turbines numerically. The turbines were configured with different solidities. The NACA 0018 profile was used for the turbine design. The study was carried out using the ANSYS® Fluent 2022R2 software, two-dimensional (2D) simulations set up constant operating conditions. Rotation speed variations have been set between 21 and 74 RPM with 10 rpm increments. Furthermore, the General Richardson extrapolation method is used for the analysis of mesh convergence, monitoring the turbine power coefficient as a convergence parameter. The numerical results show that the turbine H-Darrieus with a solidity of 1.0, a wider operating range, and lower power and torque coefficient. At low TRS, the largest solidity provided the best efficiency and the greatest self-starting capability, but it also had the smallest operating range

Richardson extrapolation method for 2D-SPP on VB mesh singularly perturbed convection–diffusion problem on a Vulanović–Bakhvalov mesh

Richardson extrapolation method for 2D-SPP on VB mesh singularly perturbed convection–diffusion problem on a Vulanović–Bakhvalov mesh

A high-order compact difference method and its Richardson extrapolation for semi-linear reaction–diffusion equations with piecewise continuous argument in diffusion term

In this paper, for the initial–boundary value problems of semi-linear reaction–diffusion equations with piecewise continuous argument in spatial derivative, we suggest Crank–Nicolson method, high-order compact difference (HOCD) method and HOCD-based Richardson extrapolation (RHOCD) method. Under the appropriate conditions, it is proved that HOCD (resp. RHOCD) method has the computational accuracy O(τ2+h4) (resp. O(τ4+h4)). This shows that RHOCD method improves the calculation accuracy of HOCD method in temporal direction. Moreover, we also analyze the stability of HOCD method and thus derive a global stability criterion of this method. Finally, with a series of numerical experiments, we further confirm the computational effectiveness and theoretical accuracy of the concerned methods.

Prediction of Wind Loadings on Offshore Wind Turbines Using CFD

Analysing the structural response of offshore wind turbines (OWTs) requires a robust method to estimate environmental loadings associated with extreme weather conditions. This study aimed to accurately predict wind loadings on a non-rotating OWT using a Computational Fluid Dynamics (CFD) code, Ansys Fluent package. The inlet velocity boundary condition was defined as a uniform inlet wind speed and then as a Normal Wind Profile (NWP). Three large wind speeds of 25 m/s, 40 m/s and 50 m/s were tested with the aid of the RANS equations and the Shear Stress Transport (SST) turbulence model. A thorough mesh convergence study was conducted for both 2D and 3D simulations, and their results were assessed using the Richardson extrapolation method. Overall, the NWP method was found to produce larger wind forces and moments in comparison with the uniform wind speed conditions (on average 52% and 63% higher, respectively), and its predictions were consistent with the estimations obtained using the simplified drag formula recommended by offshore standards (within approx. 7% difference for wind forces). This paper provided the preliminary steps towards investigating the structural integrity of OWTs under extreme weather conditions.

Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

<abstract><p>In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.</p></abstract>

A two-level nesting smoothed extended meshfree method for static and dynamic fracture mechanics analysis of orthotropic materials

This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method.

Novel Romberg approximation of the Gaussian Q function and its application over versatile κ − μ shadowed fading channel

In this paper, we propose extremely accurate, generic, exponential based approximations for the Gaussian Q function. Precisely, we make use of the Romberg numerical integration technique which utilizes the Richardson extrapolation method. This method combines two trapezoidal based approximations thereby yielding accurate yet simple approximations for the Gaussian Q function. We extensively illustrate the significance of the proposed work by comparing it with several other approximations available in the open literature. To show the application of the proposed work in communication theory, as an instance, we further apply the six-term proposed approximation over the versatile κ−μ shadowed fading channel which facilitates the error performance metrics for several wireless communication systems. In addition, the analysis is solidified with the help of computer simulations.

CFD Study on Hydrodynamic Performances of a Planing Hull

The scope of the present study is to investigate the effects of various geometrical hull features, such as tunnels, spray rails and whiskers on the hydrodynamic performance of a high-speed planing hull. The criteria being tested to emphasize the boat performance are the total drag, sinkage and trim angle. In addition, the decomposition of the resistance into viscous and wave-making resistance are taken into consideration. The study starts with a validation test against experimental data in order to accentuate the capability of the Computational Fluid Dynamics CFD simulation to accurately predict the total drag and trim angle of the initial form. This is later followed by a verification study based on the Richardson Extrapolation method with a grid- and time-step-convergence test in order to predict the numerical errors during the simulation. After establishing the simulation parameters regarding the proper grid size and time step, the comparative study takes place for five hull shapes and two whisker configurations while the boat is sailing at eight different speeds. The assessment of the hydrodynamic flow parameters is evaluated compared to the initial form in order to investigate the influence of the geometry change on the hydrodynamic performances of the boat. Validation of the numerical results showed the reliability of the CFD simulation to accurately predict the drag and trim angle of the boat, while the comparative study revealed that the total drag can be reduced by up to 9%, especially at higher speeds.

A two-level nesting smoothed extended meshfree method for dynamic fracture analysis

A two-level nesting smoothed extended meshfree method (S-XMM-N) is presented for modeling dynamic fracture behaviors in two-dimensional linear elastic solids. The extended meshfree method based on the partition of unity allows for arbitrary discontinuities, but with low computational efficiency needs improvement. Meanwhile, directly introducing the generalized smoothed strain technique into the extended meshfree method can save more computational times, and with low accuracy also improvement. The proposed method combined the best feature of both the recently developed two-level nesting smoothing meshfree method and the extended radial point interpolation method (XRPIM). On the one hand, the S-XMM-N saves much more computational time than XRPIM by using the proposed two-level nesting triangular sub-domains smoothing integration, where only fewer integration points are employed and the derivatives of shape functions do not need to be calculated for constructing the stiffness matrix. On the other hand, the computational accuracy can be ensured due to the employment of Richardson extrapolation method. The results for the static and dynamic SIFs are compared with the analytical and other numerical solutions by several benchmark fracture problems.

3D ray tracing methods across inhomogeneous refractive indices using a transmission equation

When an aircraft flies at high speed in a dense atmosphere, the airflow outside the optical window interacts to form a complex flow field structure with irregular and nonuniform refractive index distribution, which makes it difficult to accurately calculate the ray propagation path based on the refractive index field. To solve this challenge, three ray tracing implementation methods with fourth-order accuracy are proposed. By comparing with the spiral ray resolution results, it is found that the Adams method has the smallest computational error of 1.2 × 10 − 11 compared with the fourth-order Runge–Kutta method and the Richardson extrapolation method. The Adams method is the most computationally efficient in terms of computational time cost, followed by the Richardson extrapolation method, and finally the fourth-order Runge–Kutta method. The results show that the Adams method is the fastest and most accurate for different step sizes and grid volumes.

Fitted Operator Method Using Multiple Fitting Factors for Two Parameters Singularly Perturbed Parabolic Problems

In this paper, we produce ϵ , μ − uniform numerical method for a singularly perturbed parabolic differential equation with two parameters. To approximate the solution, we consider the implicit Euler method for time direction, the finite difference method for spatial direction on a uniform mesh, and the fitted operator method with multiple fitting factors. To accelerate the convergence of the method, the Richardson extrapolation method is applied. The results show that the proposed method is second-order convergent in both temporal and spatial directions. The convergence of the scheme is insensitive to the two perturbation parameters. Two model examples are considered to validate the applicability of the proposed method and produced more accurate results compared to some methods that appear in the literature. Matlab software is used to manipulate the results.

Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations

The nonlinear couple wave equations, which are extensively applied in scientific fields, such as, solid state physics, quantum mechanics, nonlinear optics, are a kind of important evolution equations. This paper is concerned with their numerical solutions via the combinations of compact difference method, Predictor–Corrector (P–C) iterative methods and Richardson extrapolation methods (REMs). Firstly, fourth-order compact difference methods are used to discrete temporal and spatial derivatives, thus forming a nonlinear fully discrete compact difference scheme. By utilizing the discrete energy analysis method and fixed point theorem, we can prove that under the condition of accepted stable criterion this scheme is conditionally convergent with an order of O(τ4+hx4+hy4) in H1-norm, and solvable. For avoiding solving the system of nonlinear algebraic equations, a P–C iterative method is introduced to save time cost and make the implementation simple. Besides, REMs are further applied to attain higher-order accurate approximate solutions. Finally, numerical findings confirm the exactness of theoretical results and efficiency of the algorithms.

Discretization Analysis of Fluid Mechanical Flow Field Grid

The design basis of the performance of special industrial fluid machinery is to obtain the distribution of air flow parameters through the solution of flow field. In the numerical solution of flow field, the discretization of computational domain grid is the key factor affecting the calculation results. In this paper, the two-dimensional axisymmetric flow field analysis model were adopted. By establishing the software interface and automatic execution script file, the grid discreteness analysis process was established. The Richardson extrapolation method was used to infer the accurate solution of the flow field, and the grid independence and convergence analysis and the grid spacing analysis of the first layer of the wall were carried out. The results showed that the numerical results gradually approached the exact solution with the increase of the total number of grids, but the correlation between the numerical results and the total number of grids decreased in this process; By changing the normal spacing of the first layer of grid on each wall and making mathematical statistical analysis, the sensitive area of grid spacing could be determined. The method established in this paper could significantly improve the overall efficiency of the fluid mechanical mesh discreteness analysis and flow field analysis, and recommended the configuration of the optimal mesh discretization mode for the subsequent simulation calculation, which was conducive to further improve the calculation accuracy and efficiency of the flow field.

A Cascadic Multigrid Algorithm on the Shishkin Mesh for a Singularly Perturbed Elliptic Problem with regular layers

A two-dimensional linear elliptic equation with regular boundary layers is considered in the unit square. It is solved by using an upwind difference scheme on the Shishkin mesh which converges uniformly with respect to a small perturbation parameter. The scheme is resolved based on an iterative method. It is known that the application of multigrid methods leads to essential reduction of arithmetical operations amount. Earlier we investigated the cascadic two-grid method with the application of Richardson extrapolation to increase accuracy of the difference scheme by an order uniform with respect to a perturbation parameter, using an interpolation formula uniform with respect to a perturbation parameter. In this paper a cascadic multigrid algorithm of the same structure is studied. We construct an extrapolation of initial guess using numerical solutions on two coarse meshes to reduce the arithmetical operations amount. The application of the Richardson extrapolation method based on numerical solutions on the last three meshes leads to increase accuracy of the difference scheme by two orders uniformly with respect to a perturbation parameter. We compare the proposed cascadic multigrid method with a multigrid method with V-cycle. The results of some numerical experiments are discussed.

Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations

Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations