High-order Method Research Articles

Higher-order multi-scale deep Ritz method (HOMS-DRM) and its convergence analysis for solving thermal transfer problems of composite materials

Higher-order multi-scale deep Ritz method (HOMS-DRM) and its convergence analysis for solving thermal transfer problems of composite materials

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

In this study, we developed a reduction-consistent phase field model for non-isothermal incompressible N-phase flows. The model is based on the high-order spectral element method. To account for thermocapillary effects, we reformulated the continuum surface force model specifically for N-phase flows. In order to enhance computational efficiency, time-independent coefficient matrices for all variables involved were reconstructed, and an unsymmetrized multifrontal LU factorization was employed to solve the linear algebraic equations, which is derived by discretization. To verify the model effectiveness in calculating surface tension and describing three-phase interfacial dynamics, we conducted several experiments, including the stationary two-droplet example, the three-phase droplet spreading, and the equilibrium morphology of double emulsion droplet. Through these experiments, both the reduction-consistency and robustness of our model were demonstrated. Moreover, we validated the proposed model's applicability to non-isothermal three-phase flows by investigating the thermocapillary migration of single droplet and two droplets of different phases. Notably, we explored the thermocapillary migration of three-phase double droplets, focusing particularly on how the encapsulation process affects overall thermocapillary motion. Our findings indicate that the interaction between the two vortices near the interfaces of the two droplets strongly influences their migration behavior.

New higher-order implict method for approximating solutions of the initial value problems

New higher-order implict method for approximating solutions of the initial value problems

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Introducing high-order response surface method for improving scour depth prediction downstream of weirs

Scour depth downstream of weirs is considered one of the most important hydraulic problems, which greatly influences the stability of weirs. Recently, artificial intelligence (AI) methods have become increasingly popular in modeling hydraulic variables, especially scour depth, because they can capture nonlinear relationships between input variables and their associated objectives. Despite their importance, these models have problems with hyperparameter tuning in scour depth modeling due to their structures, so algorithms must be used to tune the hyperparameters. Moreover, these algorithms are usually tuned by using the trial-and-error method to select the hyperparameters such as the number of hidden nodes, transfer function, and learning rate, and in this case, the main problem is overfitting during the training phase. To solve these problems, the high-order response surface method (HORSM), an improved version of the response surface method (RSM), is used as an alternative approach for the first time in this study to predict the scour depth. The HORSM model is based on high-order polynomial functions (from two to six) compared with the artificial neural network model (ANN). The findings indicate that the fifth order of the HORSM polynomial function yields the most precise predictions, with a higher coefficient of determination (R2) of 0.912 and Willmott Index (WI) of 0.972 compared to the values obtained using ANN (R2 = 0.886 and WI = 0.927). Moreover, the accuracy of the predictions is represented by a reduction of the mean square error by up to 44.17 and 29.01% compared to the classical RSM and ANN, respectively. The suggested model established an excellent correlation and accuracy with experimental values.

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

High-order residual moveout correction with global optimization in local time windows

Residual moveout (RMO) correction is significant for optimizing the common image gathers (CIGs) and migration stacked sections, which is a vital step in seismic data processing. The conventional RMO correction methods flatten the migrated gathers by inverse normal moveout and then pick up the velocity parameters, which require high labor of manual picking with low efficiency. The existing high-order RMO correction methods by scanning technology need a bispectral picking process. To address these issues, we propose a novel high-order RMO correction method based on a crosscorrelation-based objective function with global optimization in local time windows. The selected time windows covering the reflected events in the stacked section are used for parameter estimation. Then the very fast simulated annealing (VFSA) algorithm is used to simultaneously invert the two effective parameters (migration velocity and anellipticity anisotropic parameter) with the local CIGs in the time window, which has relatively high computational efficiency. The Hess model example test results show that the proposed method can effectively deal with the RMO problem in the case of anisotropic media, and the developed high-order RMO correction based on anisotropic assumption significantly improved the imaging performance compared to the conventional isotropic RMO correction methods, especially for steep structures. The application to a real 3D land data set further confirms the effectiveness and practicability of the proposed method.

A Bound-Preserving and Positivity-Preserving High-Order Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Compressible Multi-Medium Flows

A Bound-Preserving and Positivity-Preserving High-Order Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Compressible Multi-Medium Flows

Polytopal discontinuous Galerkin discretization of brain multiphysics flow dynamics

A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's β-method for the elastic wave equation and the θ-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative methods that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

Spatial evolution of the double-wave-group focusing influenced by the co- and counter-propagating current

Wave–current interaction has always been a challenging topic in fluid mechanics. The research on bimodal waves has received much more attention recently, but their evolutions influenced by underlying currents are not yet clear. This study aims to investigate the effects of co- and counter-propagating currents on spatial evolution using a fully nonlinear wave-current tank based on the high-order spectral method. The process of the wave focus is significantly shortened by the counter-propagating current, resulting in a sharper crest focus, followed by the trough focus. Concurrently, the decrease in the total envelope height and width is accelerated before wave focus and then the increase is decelerated, accompanied by a delay in the envelope profile transition from the backward-leaning to the forward-leaning. The co-propagating current exhibits the opposite phenomenon. The analysis of the spectral energy distribution aids in clarifying the variation of the envelope profile. The energy redistribution, characterized by a downshift of the frequency band, and a decreased energy distribution at the second peak, along with the slightly larger value of the root mean square frequency, indicates that the energy back-flow is obstructed by the counter-propagating current. These findings contribute to our understanding of the current effect on the focused double-wave-group, providing valuable insights for future research and applications in this field.

KFVM-WENO: A High-order Accurate Kernel-based Finite Volume Method for Compressible Hydrodynamics

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

Model-free anti-disturbance tracking control for high-order discrete-time nonlinear system based on concurrent learning extended state observer

The research objective of this paper is to address the tracking control problem of high-order discrete-time nonlinear systems in the presence of completely unknown internal uncertainties, unknown external disturbances, and unknown control input gain. To address this challenge, the paper proposes a high-order discrete-time model-free anti-disturbance tracking control method without relying on prior knowledge of system parameters. The core of the approach is the development of a data-driven concurrent learning extended state observer. The uniqueness of this observer lies in its ability to estimate unknown control input gains without the persistent excitation and ensure the convergence of the estimates. Based on the concurrent learning extended state observer, a high-order discrete-time tracking control law is designed to precisely track the reference signal, ensuring the stability of the nonlinear closed-loop system. The effectiveness of this method was verified by applying simulation verification to the heading tracking control of surface automatic vehicles.

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.