High-order Accuracy Research Articles

A Quantitative Approach for the Accurate CFD Simulation of Hover in Turbulent Flow

Time-dependent Navier–Stokes simulations have been carried out for a V22 rotor in hover using an improved, lowdissipation, HLLE ++ upwind algorithm in the OVERFLOW code. Emphasis is placed on lessons learned over the past decade regarding the effects of high-order spatial accuracy, grid resolution, and the use of detached eddy simulation in predicting the figure-of-merit, the rotor's chief performance parameter. A general quick-start procedure is described together with a statistical measure of FM convergence that reduces hover computations by fourfold, similar to computational work for forward flight. Cartesian adaptive mesh refinement is used to resolve the tip–vortex to its correct physical size. This adaptive mesh refinement in the rotor wake also revealed a complex turbulent flow with worm-like structures of various scales. These structures were numerically found a decade ago and recently observed in experiment.

Математическое моделирование прогноза урожайности сельскохозяйственных культур на основе данных полевого мониторинга и дистанционного зондирования

A mathematical model for forecasting crop yields based on field monitoring data and remote sensing of the Earth has been constructed. The model includes the following main values: vegetation indices NDVI, total solar radiation flux at the lower boundary of the atmosphere, efficiency of using photosynthetically active solar radiation, biomass respiration costs. After the parameterization of the mathematical model using observational data, the number of uncertain (calibration) coefficients of the model decreases from 8 to 2. These coefficients are determined by the method of successive approximations when comparing the results of calculations with observational data on the yield of a particular agricultural crop (ACC) in a given field. It is shown that the values of these coefficients strongly depend on the choice of optimal conditions for the growth of ACC. To reduce the yield forecast error, an approach based on the numerical integration of the total energy flux density by methods of the second and fourth orders of accuracy is proposed. When using numerical integration methods of a high order of accuracy, the yield forecast error decreases on average by 20% compared to the widely used model for calculating biomass growth, which has the first order of accuracy.

A three level linearized compact difference scheme for a fourth-order reaction-diffusion equation

A high-order accuracy finite difference scheme is investigated to solve the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A three level linearized compact finite difference scheme is proposed. Priori estimates and unique solvability are discussed in detail by the discrete energy method. The unconditional stability and convergence of the difference solution are proved. The new compact difference scheme has second-order accuracy in time and fourth-order accuracy in space in maximum norm. Numerical experiments demonstrate the accuracy, efficiency of our proposed technique.

ADER discontinuous Galerkin Material Point Method

AbstractThe first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black–Scholes model

The Black–Scholes (B–S) equation has been recently extended as a kind of tempered time-fractional B–S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B–S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.

An efficient implementation of compact third-order implicit reconstruction solver with a simple WBAP limiter for compressible flows on unstructured meshes

In this paper, we present the development of a third-order density-based solver within the OpenFOAM framework, tailored for handling compressible flows. The solver incorporates implicit variational reconstruction on three-dimensional unstructured meshes, as well as a novel technology coupling reconstruction and time integration for both steady and unsteady simulations. To address the challenge of achieving high-order accuracy for curved geometries, we introduce a new approach for curved wall boundary reconstruction, specifically designed for situations where high-order mesh information is not readily available in OpenFOAM. Furthermore, we propose a simple WBAP limiter capable of capturing shocks without necessitating the whole-domain successive limiting procedure. Numerical tests were conducted to assess the solver's performance. The results reveal that our established solver exhibits higher accuracy in smooth flow simulations, while maintaining an excellent balance between accuracy and robustness for problems involving strong shocks and other complex flow structures.

A class of ENO schemes with adaptive order for solving hyperbolic conservation laws

When solving hyperbolic conservation laws, in order to avoid spurious oscillations at discontinuities, namely the Gibbs phenomenon, essentially non-oscillatory (ENO) schemes select the smoothest stencil from several candidates of an equal size to approximate the fluxes. However, when all candidate stencils are smooth, we can use the union of the candidate stencils to construct a polynomial with higher-order accuracy. In extreme situations, all of the candidate stencils of the same size are possibly non-smooth. In view of this problem, we proposed ENO schemes with adaptive order (ENO-AO) which select a polynomial from several candidates that are reconstructed on stencils of unequal sizes. The selected polynomial is the optimal one in the sense that it predicts the cell average of the neighboring cell on either side of the target stencil with the smallest error. In this way, all the stencils containing discontinuities are automatically excluded, and the selected polynomial approximates the flux to adaptive order, ranging from the first-order all the way up to the designed high-order. When applied to piecewise smooth data, the proposed ENO-AO schemes give high-order accuracy whenever the data is smooth but avoid the Gibbs phenomenon at discontinuities. Several numerical examples of scalar conservation laws and the compressible Euler equations are carried out to test the accuracy, robustness, and efficiency of the proposed schemes.

A new type of increasingly higher order finite difference and finite volume MR-WENO schemes with adaptive linear weights for hyperbolic conservation laws

In this paper, the new fifth-order, seventh-order, and ninth-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) schemes with adaptive linear weights are presented for hyperbolic conservation laws on structured meshes. They are termed as high-order finite difference and finite volume ALW-WENO schemes. These ALW-WENO schemes only apply one small stencil and one large stencil in reconstruction processes, which could achieve the desired accuracy in the region of smoothness and non-oscillatory properties in the region of containing strong shocks. The linear weights that sum to one can be automatically adjusted to any positive numbers. This is the first time that arbitrary high-order finite difference and finite volume WENO schemes are designed by using only two unequal-sized central spatial stencils. The structure of these novel WENO schemes is simple, so it is easier for obtaining high-order accuracy and solving multi-dimensional problems in large scale engineering applications. Compared to traditional MR-WENO schemes with same order, the computational efficiency can be further improved. Some benchmark tests indicate that these new ALW-WENO schemes have good robustness and performance.

A high-order accuracy method for calculating the initial icing stage of civil aircraft structural elements

Представлен эффективный подход на основе разрывного метода Галеркина высокого порядка точности для расчета начальной стадии обледенения крыла самолета. Задача решена в приближении Эйлера для водяных капель малого размера, не оказывающих влияния на основной поток. Выписаны системы уравнений Навье-Стокса и модели Эйлера для движения водной среды и некоторые соотношения уравнений термодинамики нарастания льда. Сформулированы начальные и граничные условия. Предложена суперкомпьютерная реализация разрывного метода Галеркина для решения указанных систем уравнений. Исследована эффективность параллельной версии программы. Даны комментарии по особенностям организации процедуры расчета. Исследована точность расчета с применением схем разрывного метода Галеркина различного порядка точности. Представлены тестовые случаи обтекания цилиндра и профиля NACA0012 мелкодисперсным потоком переохлажденных капель. Проведено сопоставление расчетных и экспериментальных данных. Сделано заключение о возможности применения разработанной методологии на практике.

A free-surface particle regularization scheme based on numerical integration for particle methods

Particle shifting model plays a significant role in regularizing particle distribution in high-order accuracy Lagrangian particles methods. However, due to the incomplete neighbouring support of surface particles, the treatment of particle shifting models applied to free-surface particles still deserves attention from researchers. A new free-surface particle regularization scheme based on numerical integration (NIPS) for particle methods is proposed in this study. Specifically, high-resolution surrounding meshes are seeded outside the free surfaces to supplement the incomplete neighbouring support. Then, the contribution of the void region outside the free surface to the particle shifting vector is numerically integrated based on the surrounding meshes. The appropriate resolution and the screen scheme for constructing the surrounding meshes are optimized in this study. This scheme is easy to implement because the particle number density is only employed to screen the surrounding meshes. A typical high-order accuracy particle method, the LSMPS method is taken to evaluate the proposed scheme in this study. Numerical examples demonstrated that the proposed regularization method can keep uniform distribution of surface particles, thus the instability due to heterogeneous surface particles distribution can be effectively avoided.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

High-order supplementary variable methods for thermodynamically consistent partial differential equations

In recent years, there has been growing interest in developing numerical methods to retain consistency between the partial differential equation system and its deduced equations (e.g., the thermodynamically consistent model and its energy law) after numerical approximations. A supplementary variable method (SVM) of second-order was recently introduced in [1,2] to preserve thermodynamic consistency for thermodynamically consistent models. In this paper, we present a new class of high-order supplementary variable methods. These methods consist of two parts: firstly, the original model is discretized using a high-order prediction method, and secondly, the SVM reformulated system is approximated by a high-order correction method. The newly developed schemes are rigorously proven to preserve high-order accuracy with increased stability through the use of a stabilizer. Additionally, the schemes inherently maintain thermodynamic consistency for thermodynamically consistent models, retaining the original energy law at the discrete level. To assess the high-order supplementary variable methods, we apply them to two thermodynamically consistent partial differential equation systems: the functionalized Cahn–Hilliard model and the ternary Cahn–Hilliard model. We then provide several benchmark examples to demonstrate the accuracy, stability, and efficiency of the newly developed high-order numerical methods.

Stabilized isogeometric collocation methods for scalar transport and incompressible fluid flow

In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation schemes address spurious oscillations that can arise from advection and pressure instabilities. Numerical examples for the advection–diffusion equation, Stokes equations, and incompressible Navier–Stokes equations show the effectiveness of the proposed stabilized schemes while maintaining the high-order convergence rates and accuracy of standard isogeometric collocation on smooth problems.

HOW-MHD: A High-order WENO-based Magnetohydrodynamic Code with a High-order Constrained Transport Algorithm for Astrophysical Applications

Due to the prevalence of magnetic fields in astrophysical environments, magnetohydrodynamic (MHD) simulation has become a basic tool for studying astrophysical fluid dynamics. To further advance the precision of MHD simulations, we have developed a new simulation code that solves ideal adiabatic or isothermal MHD equations with high-order accuracy. The code is based on the finite-difference weighted essentially nonoscillatory (WENO) scheme and the strong stability-preserving Runge–Kutta (SSPRK) method. Most of all, the code implements a newly developed, high-order constrained transport (CT) algorithm for the divergence-free constraint of magnetic fields, completing its high-order competence. In this paper, we present the version in Cartesian coordinates, which includes a fifth-order WENO and a fourth-order five-stage SSPRK, along with extensive tests. With the new CT algorithm, fifth-order accuracy is achieved in convergence tests involving the damping of MHD waves in 3D space. And substantially improved results are obtained in magnetic loop advection and magnetic reconnection tests, indicating a reduction in numerical diffusivity. In addition, the reliability and robustness of the code, along with its high accuracy, are demonstrated through several tests involving shocks and complex flows. Furthermore, tests of turbulent flows reveal the advantages of high-order accuracy and show that the adiabatic and isothermal codes have similar accuracy. With its high-order accuracy, our new code would provide a valuable tool for studying a wide range of astrophysical phenomena that involve MHD processes.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Diagonalized Legendre spectral method for second-order eigenvalue problems

Diagonalized Legendre spectral methods for solving second-order eigenvalue problems are proposed. A new class of basis functions are constructed by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products, and lead to diagonal systems for second-order eigenvalue problems with constant coefficients. Numerical experiments show that the suggested approaches possess high-order accuracy and greatly improve the efficiency. Compared with the classical methods, the new methods simplify the calculation process, reduce the amount of calculation, and can effectively avoid the curse of dimensionality of high-dimensional eigenvalue problems.

A moment-of-fluid method for resolving filamentary structures using a symmetric multi-material approach

Multiphase flows have implications in many areas of engineering. The moment-of-fluid (MOF) method is an interface capturing method using both volume fraction and centroid within a cell for interface reconstruction. A symmetric approach to reconstruct thin structures is presented. Also called filaments, these subcell characteristics involve multi-material reconstruction. In addition, a new optimisation algorithm is presented using a bisection method without any necessary initial condition. Using a Lagrangian approach for dynamic cases, no restrictions are imposed on timestep. The new method is validated using several benchmark cases that have been studied extensively in the literature. A near quadratic order of convergence and high accuracy is achieved while maintaining an acceptable runtime.

A New Approach to Increase the Accuracy of a Numerical Scheme Using Modified Partial Differential Equation : Application to Meshless Methods

Meshless methods are gaining popularity for numerically solving the governing equations of Fluid Dynamics primarily because they do not require the generation of grid around the given geometry in the computational domain. They just require a point distribution with a proper connectivity, which is extremely simple to generate as compared to the grid around complex geometries. However, obtaining a higher order accurate numerical solution is difficult in case of meshless methods as it requires either a solution of a larger system of linear algebraic equations, or defect correction method coupled with inner iterations. This method becomes too much complicated if still further higher order accuracy is desired. Here an alternate method based on Modified Partial Differential Equations has been used to obtain higher order accurate solution using the meshless methods. The basic idea behind the new approach is to explicitly remove the truncation error from the Modified Partial Differential Equation (formed from the governing equations) and numerically solve the resulting equation with the meshless technique. This technique has been successfully applied for the simulation of subsonic, weak transonic, strong transonic and supersonic flow past NACA0012 aerofoil. A good match of the numerical results with the standard test results of AGARD/GAMM workshop has been.

High-order corrected trapezoidal rules for a class of singular integrals

We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High-order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the implicit boundary integral formulation of surface integrals involving the Laplace layer kernels.