High-order Accuracy Research Articles

Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations

It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems.

A WENO-Based Upwind Rotated Lattice Boltzmann Flux Solver with Lower Numerical Dissipation for Simulating Compressible Flows with Contact Discontinuities and Strong Shock Waves

This paper presents a WENO-based upwind rotated lattice Boltzmann flux solver (WENO-URLBFS) in the finite difference framework for simulating compressible flows with contact discontinuities and strong shock waves. In the method, the original rotating lattice Boltzmann flux solver is improved by applying the theoretical solution of the Euler equation in the tangential direction of the cell interface to reconstruct the tangential flux so that the numerical dissipation can be reduced. The fluxes at each interface are evaluated using a weighted summation of lattice Boltzmann solutions in two local perpendicular directions decomposed from the direction vector so that the stability performance can be improved. To achieve high-order accuracy, both fifth and seventh-order WENO reconstructions of the flow variables in the characteristic spaces are carried out. The order accuracy of the WENO-URLBFS is evaluated and compared with the traditional Lax–Friedrichs scheme, Roe scheme, and the LBFS by simulating the advection of the density disturbance problem. It is shown that the fifth and seventh-order accuracy can be achieved by all considered flux-evaluation schemes, and the present WENO-URLBFS has the lowest numerical dissipation. The performance of the WENO-URLBFS is further examined by simulating several 1D and 2D examples, including shock tube problems, Shu–Osher problems, blast wave problems, double Mach reflections, 2D Riemann problems, K-H instability problems, and High Mach number astrophysical jets. Good agreements with published data have been achieved quantitatively. Moreover, complex flow structures, including shock waves and contact discontinuities, are successfully captured. The present WENO-URLBFS scheme seems to present an effective numerical tool with high-order accuracy, lower numerical dissipation, and strong robustness for simulating challenging compressible flow problems.

Uncertainty quantification using simulation output: batching as an inferential device

ABSTRACT We present batching as an omnibus device for uncertainty quantification using simulation output. We consider the classical context of a simulationist performing uncertainty quantification on an estimator θ n (of an unknown fixed quantity θ ) using only the output data ( Y 1 , Y 2 , … , Y n ) gathered from a simulation. By uncertainty quantification, we mean approximating the sampling distribution of the error θ n − θ toward: (A) estimating an “assessment” functional ψ , eg, bias, variance, or quantile; or (B) constructing a ( 1 − α ) -confidence region on θ . We argue that batching is a remarkably simple and effective device, and is especially suited for handling dependent output data which frequently appears in simulation contexts. We demonstrate that with appropriately chosen batch numbers and extent of overlap, batching retains bootstrap’s attractive theoretical properties of strong consistency and higher-order accuracy. Our extensive numerical experience confirms theoretical insight, especially about the effects of batch size and overlap.

Efficient One-Step Second Derivative Block Numerical Scheme for Solution of first-Order Integro-differential Equations

In this work, a new class of one-step second derivative block hybrid numerical scheme is developed for the numerical solution of first-order integro-differential equations with the aid of Simpson's 1/3 quadrature method. The collocation techniques is used for the derivation of this new block method which gives high order of accuracy with very low error constants and its zero stable and convergent. The results from the numerical solution of the examples drawn from literature shows its efficiency in terms of the small errors obtained.

Numerical stability analysis of shock-capturing methods for strong shocks II: high-order finite-volume schemes

The shock instability problem commonly arises in flow simulations involving strong shocks, particularly when employing high-order shock-capturing schemes, limiting their application in hypersonic flow simulations. In the current study, the shock instability problem of the fifth-order finite-volume WENO scheme is investigated. To this end, the matrix stability analysis method for the fifth-order scheme is established. Such an analysis method is capable of providing quantitative insights into the stability of shock-capturing and helping elucidate the mechanism causing shock instabilities. Results from the stability analysis and numerical experiments reveal that even dissipative solvers, which are generally believed to be capable of capturing strong shocks stably, also suffer from the shock instability problem when the spatial accuracy is increased to the fifth-order. Further investigation indicates that this is because the quadratic polynomial degenerated from the WENO reconstruction is not applicable for reconstructing variables on the faces near the numerical shock structure. Moreover, the shock instability problem of the fifth-order scheme is demonstrated to be a multidimensional coupled problem. To capture strong shocks stably, it is necessary to have sufficient dissipation on both transverse faces and normal faces in the vicinity of strong shocks. The spatial location of the perturbation leading to instability is also clarified by the proposed matrix analysis and numerical experiments, revealing that the instability arises from the numerical shock structure for the fifth-order scheme. Additionally, the local characteristic decomposition is demonstrated to be helpful to mitigate shock instabilities in high-order cases. Based on these conclusions and the core idea of the MR-WENO scheme, a shock-stable high-order scheme, the MR-WENO-SD scheme, is proposed. Equipped with the hybrid HLLC/HLL solver, this scheme is capable of capturing strong shocks stably while maintaining high-order accuracy. A series of numerical experiments demonstrate its stability and resolution capabilities.

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

CESE Schemes for Solar Wind Plasma MHD Dynamics

Magnetohydrodynamic (MHD) numerical simulation has emerged as a pivotal tool in space physics research, witnessing significant advancements. This methodology offers invaluable insights into diverse space physical phenomena based on solving the fundamental MHD equations. Various numerical methods are utilized to approximate the MHD equations. Among these, the space–time conservation element and solution element (CESE) method stands out as an effective computational approach. Unlike traditional numerical schemes, the CESE method significantly enhances accuracy, even at the same base point. The concurrent discretization of space and time for conserved variables inherently achieves higher-order accuracy in both dimensions, without the need for intricate higher-order time discretization processes, which are often challenging in other methods. Additionally, this scheme can be readily extended to multidimensional cases, without relying on operator splitting or direction alternation. This paper primarily delves into the remarkable progress of CESE MHD models and their applications in studying solar wind, solar eruption activities, and the Earth’s magnetosphere. We aim to illuminate potential avenues for future solar–interplanetary CESE MHD models and their applications. Furthermore, we hope that the discussions presented in this review will spark new research endeavors in this dynamic field.

Simulation of Scalar Wave Propagation with High-Order Temporal and Spatial Accuracy by a New Multi-Axial Staggered-Grid Finite-Difference Scheme

ABSTRACT Staggered-grid finite-difference (SFD) stencils are extensively applied for scalar wavefield simulations and inversions in seismology because of their easy implementation and effectiveness of propagating the wave in heterogeneous media. The conventional SFD (CSFD) stencil adopts second-order temporal and high-order spatial finite-difference operators to approximate the partial derivatives inside the wave equation. The spatial SFD operator only adopts grid points along one orthogonally axial direction to approximate the spatial partial derivative along that direction. Therefore, increasing the number of grid points along the axis will not improve the temporal accuracy. To simultaneously enhance the temporal and spatial accuracy, we propose a new multi-axial SFD (MASFD) stencil, which consists of grid points along three directions for each partial derivative in space. The MASFD weightings (coefficients) are derived by preserving the dispersion relation of the scalar wave in the frequency–wavenumber domain. We prove that increasing the number of the grid points of the new stencil can simultaneously reach high-order accuracy in time and space. The performance of the new MASFD scheme is compared with the CSFD schemes by quantitative dispersion analyses, stability analyses, and numerical examples. Our comprehensive comparisons demonstrate that the MASFD scheme can be more accurate than the CSFD ones because of improved temporal accuracy. Under comparable accuracy, the MASFD scheme can be more efficient than the CSFD ones because the MASFD scheme can adopt larger time steps to perform stable wave extrapolation.

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.

Evaluation of flexural strength of 3D-Printed nylon with carbon reinforcement: An experimental validation using ANN

This study investigates the flexural strength of 3D-printed nylon-carbon reinforced composite specimens, highlighting the impact of infill density and layer height on mechanical performance. The findings indicate that a printing layer height of 0.10 mm with 100 % infill density exhibits the highest flexural strength, supporting a maximum load of 127 N, compared to 76.7 N at 50 % infill density. Microstructural study has clearly illustrated the structural distortion, revealing that a rise in layer height correlates with an escalation in structural distortion. An Artificial Neural Network (ANN) model is thus utilized to achieve high predictive accuracy in order to predict flexural behaviour. R-values above 0.98 are obtained across training, validation, and test datasets, indicating that ANN-based modelling may be able to facilitate quick optimization of 3D printing parameters for high-performance applications. These findings establish carbon-reinforced nylon as a formidable competitor for use in industries such as aerospace and automotive, where strength and durability are important.

Positivity and bound preserving well-balanced high order compact finite difference scheme for Ripa and pollutant transport model

We construct a fourth-order accurate compact finite difference scheme that is well-balanced, positivity-preserving of water height, and bound-preserving of temperature for Ripa and concentration for pollutant transport systems. The proposed scheme preserves the still-water steady state and the positivity of water height. It also maintains concentration bounds for pollutants across nonflat bottom topographies, regardless of the presence of a pollutant source. Our approach incorporates water height and pollutant concentration constraints within the same discretization, utilizing weak monotonicity and a simple bound-preserving limiter while preserving the well-balanced property. Through extensive numerical simulations encompassing Ripa and pollutant transport models, we demonstrate the effectiveness of our method, verifying its well-balanced property, high-order accuracy, positivity-preserving, and bound-preserving capabilities.

A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high-order accurate.

Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration

This article aims at presenting a new local subcell monolithic Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme solving system of conservation laws on 2D unstructured grids. This is known that DG method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, a convex blending of high-order DG and first-order FV schemes will be locally performed, at the subcell scale, where it is needed. To this end, by means of the theory developed in [58,59], we first recall that it is possible to rewrite DG scheme as a subcell FV method, defined on a subgrid, provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the subcell monolithic DG/FV method will be defined as follows: to each face of each subcell we will assign two fluxes, a 1st-order FV one and a high-order reconstructed one, that in the end will be blended in a convex way. The goal is then to determine, through analysis, optimal blending coefficients to achieve the desired properties. Numerical results on various type problems will be presented to assess the very good performance of the design method.A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: is this possible through this monolithic framework to ensure any entropy stability? What do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed while aiming for high-order accuracy?

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes II: Unsteady flows

For the simulations of unsteady flow, the global time step becomes really small with a large variation of local cell size. In this paper, an implicit high-order gas-kinetic scheme (HGKS) is developed to alleviate the restrictions on the time step for unsteady simulations. In order to improve the efficiency and keep the high-order accuracy, a two-stage third-order implicit time-accurate discretization is proposed. In each stage, an artificial steady solution is obtained for the implicit system with the pseudo-time iteration. In the iteration, the classical implicit methods are adopted to solve the nonlinear system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and generalized minimum residual (GMRES) methods. To achieve the spatial accuracy, the HGKSs with both non-compact and compact reconstructions are constructed. For the non-compact scheme, the weighted essentially non-oscillatory (WENO) reconstruction is used. For the compact one, the Hermite WENO (HWENO) reconstruction is adopted due to the updates of both cell-averaged flow variables and their derivatives. The expected third-order temporal accuracy is achieved with the two-stage temporal discretization. For the smooth flow, only a single artificial iteration is needed. For uniform meshes, the efficiency of the current implicit method improves significantly in comparison with the explicit one. For the flow with discontinuities, compared with the well-known Crank-Nicholson method, the spurious oscillations in the current schemes are well suppressed. The increase of the artificial iteration steps introduces extra reconstructions associating with a reduction of the computational efficiency. Overall, the current implicit method leads to an improvement in efficiency over the explicit one in the cases with a large variation of mesh size. Meanwhile, for the cases with strong discontinuities on a uniform mesh, the efficiency of the current method is comparable with that of the explicit scheme.

An energy stable high-order cut cell discontinuous Galerkin method with state redistribution for wave propagation

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to “cut” a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani in [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is L2 energy stable under arbitrary quadrature. We prove that state redistribution can be added to a provably L2 energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's L2 stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, L2 stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.

Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

Spectral-Galerkin methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop two numerical methods, the Legendre-Galerkin method and the generalized Log orthogonal functions Galerkin method for numerically solving the fully nonlinear Monge-Ampère equation. Both methods are constructed based on the vanishing moment approach. To address both solution stability and computational efficiency, we propose a multiple-level framework for resolving discretization schemes. The mathematical justifications of the new approaches and the error estimates for the Legendre-Galerkin method are established. Numerical experiments validate the accuracy of our methods, and a comparative experiment demonstrates the advantage of Log orthogonal functions for problems with corner singularities. The results highlight that our methods have high-order accuracy and small computational cost.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

Analysis of the Effectiveness of Using a High-Accuracy Numerical Algorithm in the Simulation of Two-Phase Flow Problems

The continuous technical development of weapons and ammunition, as well as the growing requirements placed on new equipment, require the use of modern tools for their designing and testing. The analysis of the operation of powder propellant systems, which is the focus of interior ballistics, requires particularly specialised tools due to the complexity of the phenomena occurring during a shot. The development of numerical methods and the increase in the computing power of modern computers have enabled the modelling and simulation of highly complex physical problems and the development of numerical schemes with a theoretically arbitrary high order of accuracy. In this study, an attempt was made to develop a one-dimensional model of two-phase flows with a high order of solution accuracy in time, based on the finite volume method. The use of a two-phase flow model in the context of interior ballistics allows, among other things, the analysis of wave phenomena occurring during firing, which is crucial for safety reasons, especially in tank guns. The developed algorithm for solving two-phase systems was checked using recognised test problems for this type of model. The results of the work will be used to develop a numerical model to solve the main problem of interior ballistics.