High-order Accuracy Research Articles

An efficient pure meshless method for phase separation dominated by time-fractional Cahn–Hilliard equations

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution.

High-order, unconditionally maximum-principle preserving finite element method for the Allen–Cahn equation

In this paper, based on the mass-lumping finite element space discretization, we incorporate the integrating factor Runge–Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen–Cahn equation and its conservative forms. The proposed methods are linear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite element method.

High-order Coverage of Smoothed Bayesian Bootstrap Intervals for Population Quantiles

We characterize the high-order coverage accuracy of smoothed and unsmoothed Bayesian bootstrap confidence intervals for population quantiles. Although the original (Rubin 1981) unsmoothed intervals have the same O(n−1/2) coverage error as the standard empirical bootstrap, the smoothed Bayesian bootstrap of Banks (1988) has much smaller O(n−3/2[log(n)]3) coverage error and is exact in special cases, without requiring any smoothing parameter. It automatically removes an error term of order 1/n that other approaches need to explicitly correct for. This motivates further study of the smoothed Bayesian bootstrap in more complex settings and models.

An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

Nonlinear improvements of quasi-interpolanting splines to approximate piecewise smooth functions

Quasi-interpolation based on B-spline approximation methods are used in numerous applications. However, we observe that the Gibbs phenomenon appears when approximating near discontinuities. We present nonlinear modifications, based on weighted essentially non-oscillatory (WENO) techniques, of well-known quasi-interpolation methods to avoid this phenomena near discontinuities and, at the same time, maintain the high-order accuracy in smooth regions.

Extensions of Active Flux to arbitrary order of accuracy

Active Flux is a recently developed numerical method for hyperbolic conservation laws. Its classical degrees of freedom are cell averages and point values at cell interfaces. These latter are shared between adjacent cells, leading to a globally continuous reconstruction. The update of the point values includes upwinding, but without solving a Riemann Problem. The update of the cell average requires a flux at the cell interface, which can be immediately obtained using the point values. This paper explores different extensions of Active Flux to arbitrarily high order of accuracy, while maintaining the idea of global continuity. We propose to either increase the stencil while keeping the same degrees of freedom, or to increase the number of point values, or to include higher moments as new degrees of freedom. These extensions have different properties, and reflect different views upon the relation of Active Flux to the families of Finite Volume, Finite Difference and Finite Element methods.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

Hermite Interpolation Approach to High-Order Approximation of Heat Equations


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

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the discrete ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL). Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

A High-Order Compact Gas-Kinetic Scheme in a Rotating Coordinates Frame and on Sliding Mesh

This paper extends the high-order compact gas-kinetic scheme (CGKS) to flow simulations on a rotating coordinate frame. The kinetic equation with the inclusion of centrifugal and Coriolis accelerations is used in the construction of the scheme. With the updates of both cell-averaged flow variables and their gradients in the rotating and stationary domains, a third-order compact reconstruction is developed with sliding interface between them. To capture shock waves and complicated wave interactions, the HWENO-type non-linear reconstruction and gradient compression factors are incorporated in the scheme. For achieving high-order time accuracy, based on the time-accurate flux function the multi-stage multi-derivative time stepping method is implemented in the scheme for the fourth-order time accuracy with two stages. The CGKS is validated in cases from subsonic acoustic wave propagation to high Mach number shock interaction. The compact scheme achieves high-order accuracy and remarkable robustness.

Positivity-Preserving Lax–Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models

The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor–corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

Solving Navier–Stokes with mimetic operators

We present a new scheme for solving Navier–Stokes equations using mimetic difference operators. These operators can be constructed to high orders of accuracy and maintain the physical properties of the problem under consideration. We demonstrate the effectiveness of our scheme by modeling a lock release in 3D Cartesian coordinates, then extend our techniques to 3D curvilinear grids. The resulting scheme allows for simple and efficient computation of fluid processes on curvilinear grids, which allows us to solve problems in more complex regions while minimizing the restrictions of finite difference methods.

A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

We investigate the Benjamin–Feir (or modulational) instability of Stokes waves, i.e. small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Bertiet al.(Full description of Benjamin–Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Bertiet al.(Benjamin–Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin–Feir instability and for the parametrization of the Benjamin–Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin–Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

Analyzing thermal radiation in cylindrical anisotropic scattering medium with element differential method

The spontaneous radiation combustion diagnosis technology has high requirements for obtaining the radiation intensity of highly directional resolution in a combustion system. Taking advantage of the numerical stability and easy implementation of the element differential method, this paper constructs a radiation model that can achieve the radiation intensity of spatial and angular high-resolution in cylindrical anisotropic scattering medium. In analyzing the radiation model, the radiation intensity is discretized in three dimensions. An upwind scheme is proposed to suppress the numerical oscillation of the strong convection characteristics of the radiation transfer equation. The double-layer node algorithm is used to capture the strong discontinuous singularities at the radiation boundary. The comparison with the analytical solution shows that the radiation model based on the element differential method can achieve the high-resolution description of radiation intensity with high-order accuracy. The accuracy and validity of the radiation model are verified through comparing with the results on the Monte Carlo method. The further description of the three-dimensional distribution of radiation intensity in angle and space proves that the up-wind scheme can effectively suppress numerical oscillation and realize stable calculation.

English

In this paper, two numerical integration methods for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs), namely “Third Order One Step Scheme (TOOSS) and Second Order One Step Scheme (SOOSS)” have been considered. The order of convergence, consistency and the stability properties of the schemes have been investigated. From the analyses, it is observed that SOOSS and TOOSS have second order convergence and third order convergence, respectively. It is also observed that both numerical integration methods are consistent and stable. Moreover, three IVPs of stiff differential equations were solved to examine the performance of SOOSS and TOOSS in terms of absolute relative errors. Hence, the numerical results show that TOOSS performs better than SOOSS because of its higher order of accuracy. Key words: Accuracy, consistency, convergence, final absolute relative error, stability region, stiff differential equation.

Spectral difference method with a posteriori limiting: application to the Euler equations in one and two space dimensions

ABSTRACTWe present a new numerical scheme which combines the spectral difference (SD) method up to arbitrary high order with a-posteriori limiting using the classical MUSCL-Hancock scheme as fallback scheme. It delivers very accurate solutions in smooth regions of the flow while capturing sharp discontinuities without spurious oscillations. We exploit the strict equivalence between the SD scheme and a finite-volume scheme based on the SD control volumes to enable a straightforward limiting strategy. At the end of each stage of our high-order time-integration ADER (Arbitrary high order using Derivatives) scheme, we check if the high-order solution is admissible under a number of numerical and physical criteria. If not, we replace the high-order fluxes of the troubled cells by fluxes from our robust second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) fallback scheme. We apply our method to a suite of test problems for the one-dimensional and two-dimensional Euler equations. We demonstrate that this combination of SD and ADER provides a virtually arbitrary high order of accuracy while at the same time preserving good sub-element shock capturing capabilities.

An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations

AbstractAn oscillation free local discontinuous Galerkin (OFLDG) method is proposed for solving nonlinear degenerate parabolic equations. The damping terms are added to the original LDG scheme to control the spurious oscillations when solutions have a large gradient. The L2‐stability and optimal priori error estimates for the semi‐discrete scheme in one‐ and multi‐dimensions are established. The numerical experiments demonstrate that the proposed method maintains the high order accuracy and controls the spurious oscillations well.

Studies on modified limited-memory BFGS method in full waveform inversion

Full waveform inversion (FWI) is a non-linear optimization problem based on full-wavefield modeling to obtain quantitative information of subsurface structure by minimizing the difference between the observed seismic data and the predicted wavefield. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method is an effective quasi-Newton method in FWI due to its high inversion efficiency with low calculation and storage requirements. Like other conventional quasi-Newton methods, the approximation of the Hessian matrix in the L-BFGS method satisfies the quasi-Newton equation, which only exploits the gradient and model information while the available objective function value is neglected. The modified quasi-Newton equation considers the gradient, model, and objective function information together. Theoretical analysis reveals that the modified quasi-Newton equation is superior to the conventional quasi-Newton equation as it achieves higher-order accuracy in approximating the Hessian matrix. The modified L-BFGS method can be obtained by using the modified quasi-Newton equation to modify the L-BFGS method. This modification improves the accuracy of the Hessian matrix approximation with little increase of calculation for each iteration. We incorporate the modified L-BFGS method into FWI, numerical results show that the modified L-BFGS method has a higher convergence rate, achieves better inversion results, and has stronger anti-noise ability than the conventional L-BFGS method.

Three-dimensional elastic reverse-time migration using a high-order temporal and spatial staggered-grid finite-difference scheme

Three-dimensional (3D) elastic reverse-time migration (ERTM) can image the subsurface 3D seismic structures, and it is an important tool for the Earth’s interior imaging. A common simulation kernel used in 3D ERTM is the current staggered-grid finite-difference (SGFD) method of the first-order elastic wave equation. However, the mere second-order accuracy in time of the current SGFD method can bring non-negligible time dispersion, which reduces the simulation accuracy and further leads to the distortion of the imaging results. This paper proposes a vector-based 3D ERTM using the high-order accuracy SGFD method in time to obtain high-accuracy images. This approach is a new high-resolution ERTM workflow that improves the imaging accuracy of conventional ERTM from numerical simulation. The proposed ERTM workflow is established on a quasi-stress–velocity wave equation and its vector wavefield decomposition form. Advanced SGFD schemes and their corresponding coefficients with fourth-order temporal accuracy solve the quasi-linear wave equation system. The normalized dot product imaging condition produces high-quality images using high-accuracy vector wavefields solved using the SGFD method. Through the numerical examples, we test the simulation efficiency and analyze how temporal accuracy in numerical simulations affects migration imaging quality. We include that the proposed method obtains highly accurate images.