Spatial Mesh Research Articles

Higher order method based on the coupling of local discontinuous Galerkin method with multistep implicit-explicit time-marching for the micropolar Navier–Stokes equations

In this paper, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal multistep implicit-explicit (IMEX) evolution for the micropolar Navier–Stokes equations (MNSE) is adopted to construct an efficient fully discrete method. First, the multistep IMEX-LDG methods are constructed up to the third order, which have small computational scale and facilitate implementation to higher orders. Second, the unconditional stability of the fully discrete method on Cartesian grids is derived theoretically, meaning that the temporal step size τ is upper bounded by a positive constant independent of the spatial mesh size h. Last, numerical results for nonlinear MNSE with different boundary conditions are presented, confirming the theoretical validity and effectiveness of the method.

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Effects of OpenCL-Based Parallelization Methods on Explicit Numerical Methods to Solve the Heat Equation

In recent years, the need for high-performance computing solutions has increased due to the growing complexity of computational tasks. The use of parallel processing techniques has become essential to address this demand. In this study, an Open Computing Language (OpenCL)-based parallelization algorithm is implemented for the Constant Neighbors (CNe) and CNe with Predictor–Corrector (CpC) numerical methods, which are recently developed explicit and stable numerical algorithms to solve the heat conduction equation. The CPU time and error rate performance of these two methods are compared with the sequential implementation and Euler’s explicit method. The results demonstrate that the parallel version’s CPU time remains nearly constant under the examined circumstances, regardless of the number of spatial mesh points. This leads to a remarkable speed advantage over the sequential version for larger data point counts. Furthermore, the impact of the number of timesteps on the crossover point where the parallel version becomes faster than the sequential one is investigated.

Space–time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation

The space–time adaptive ADER–DG finite element method with LST–DG predictor and a posteriori sub–cell ADER–WENO finite–volume limiting was used for simulation of multidimensional reacting flows with detonation waves. The presented numerical method does not use any ideas of splitting or fractional time steps methods. The modification of the LST–DG predictor has been developed, based on a local partition of the time step in cells in which strong reactivity of the medium is observed. This approach made it possible to obtain solutions to classical problems of flows with detonation waves and strong stiffness, without significantly decreasing the time step. The results obtained show the very high applicability and efficiency of using the ADER–DG–PN method with a posteriori sub–cell limiting for simulating reactive flows with detonation waves. The numerical solution shows the correct formation and propagation of ZND detonation waves. The structure of detonation waves is resolved by this numerical method with subcell resolution even on coarse spatial meshes. The smooth components of the numerical solution are correctly and very accurately reproduced by the numerical method. Non–physical artifacts of the numerical solution, typical for problems with detonation waves, such as the propagation of non–physical shock waves and weak detonation fronts ahead of the main detonation front, did not arise in the results obtained. The results of simulating rather complex problems associated with the propagation of detonation waves in significantly inhomogeneous domains are presented, which show that all the main features of detonation flows are correctly reproduced by this numerical method. It can be concluded that the space–time adaptive ADER–DG–PN method with LST–DG predictor and a posteriori sub–cell ADER–WENO finite–volume limiting is perfectly applicable to simulating fairly complex reacting flows with detonation waves.

Optimizing the selective laser melting molding process for TC4 spatial mesh structures

Optimizing the selective laser melting molding process for TC4 spatial mesh structures

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented.

An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray–Scott system

This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order α∈(1,2) in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form u=0 in R∖Ω. The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order α: from self-replication to standing waves and from travelling waves to self-replication.

Accelerating particle-in-cell kinetic plasma simulations via reduced-order modeling of space-charge dynamics using dynamic mode decomposition.

We present a data-driven reduced-order modeling of the space-charge dynamics for electromagnetic particle-in-cell (EMPIC) plasma simulations based on dynamic mode decomposition (DMD). The dynamics of the charged particles in kinetic plasma simulations such as EMPIC is manifested through the plasma current density defined along the edges of the spatial mesh. We showcase the efficacy of DMD in modeling the time evolution of current density through a low-dimensional feature space. Not only do such DMD-based predictive reduced-order models help accelerate EMPIC simulations, they also have the potential to facilitate investigative analysis and control applications. We demonstrate the proposed DMD-EMPIC scheme for reduced-order modeling of current density and speedup in EMPIC simulations involving electron beam under the influence of magnetic field, virtual cathode oscillations, and backward wave oscillator.

A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equation

The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolsondiscretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions. Model examples are provided to validate the practicality of the proposed numerical scheme.

COLD INHIBITION OF BIOACTIVITY OF PROBIOTIC MYCOBACTERIA AND AEROCOCCI IN THE MICROBIOME

The organism of farm animals is constantly under the influence of various factors of the external environment, and one of the negative factors of violation of the technology of keeping and creation of unsatisfactory conditions of existence with deterioration of well-being is cold injury. Mostly it manifests itself as a permanent effect of positive temperatures well below the physiological norm or periodic fluctuations of an uncomfortable temperature. Such a stress factor of low intensity, but permanent over a long period of time, causes negative pathophysiological changes in the macroorganism with the development of immune-depression, metabolic syndrome, perversions of the qualitative composition of the microbiome and, as a result, disruption of native physiological processes of stimulation of cellular mechanisms of nonspecific innate immunity, induced by immune-biological mechanisms of the functioning of neutrophil granulocytes. The negative effect of low-intensity permanent cold stress was studied on a model object - murchakаs, which were kept for two weeks at a temperature of 4-6 °C under conditions of light and movement restriction. Before the start of the experiment, initial microbiological studies of the microbiome of the large intestine were conducted using generally accepted methods. Resident indigenous probiotic prokaryotes - M. vaccae & A. viridans, which possessed typical species characteristics, were isolated. Pathogenic variants of enterobacteria were not isolated, isolated cultures of E. coli were apathogenic and belonged to the usual normal flora. A two-week cold injury resulted in the disappearance of the indigenous probiotic microbiota M. vaccae & A. viridans, which are indicator prokaryotes of the physiological well-being of the macroorganism, from the microbiome. At the same time, the number of potentially pathogenic enteral microorganisms, coccal and rod-shaped, increased. This was correlated with a decrease in the phagocytic activity of native neutrophil granulocytes and their preformed filamentous spatial nucleoprotective mesh structures, which inhibit the de-fragmentary function in relation to genetically non-syngenic objects.

Error Propagation and Control in 2D and 3D Hybrid Seismic Wave Simulations for Box Tomography

ABSTRACT To enhance the local resolution of global waveform tomography models, particularly in areas of interest within the Earth’s deep structures, a higher resolution localized tomography approach (referred to as “box tomography”) is crucial for a more detailed understanding of the Earth’s internal structure and geodynamics. Because the small-scale features targeted by box tomography are finer than those in global reference models, distinct spatial meshes are necessary for global and local (hybrid) forward simulations. Within the spectral element method (SEM) framework, we employ the intrinsic Lagrangian spatial interpolation to compute and store hybrid inputs (displacement/potential) in the global numerical simulation. These hybrid inputs are subsequently imposed into the localized domain during the iterative box tomography. However, inaccurate spatial Lagrange interpolation can lead to imprecise hybrid inputs, and this error can propagate from the global simulation to the hybrid simulation. It is essential to quantitatively analyze this error propagation and control it to ensure the credibility of box tomography. We introduce a unique spatial window function into the conventional “direct discrete differentiation” hybrid method. When the local mesh and structure align with those in the global simulation, the synthetic hybrid waveforms match the global ones, serving as a reference for quantitatively assessing error propagation stemming from changes in the local spatial mesh during hybrid simulation. Significantly, the relative waveform error arising due to spatial Lagrange interpolation is around 5% when employing the traditional SEM with five Gauss–Lobatto–Legendre points per minimum wavelength in the 3D global simulation through SPECFEM3D_GLOBE. Ultimately, we achieve hybrid waveforms with an accuracy of about 1.5% by increasing the spectral elements by about 1.5 times in the standard global simulation.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation

In this paper, a novel linearized second-order energy-stable fully discrete scheme for the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation is introduced, along with a superconvergence analysis of the conforming finite element method (FEM). Through skillful decomposition of the nonlinear term u∇⋅u, a new linearized second-order fully discrete scheme is developed. This scheme requires fewer iterations and exhibits higher computational efficiency compared to the nonlinear approach. Furthermore, it maintains energy stability, enabling direct numerical solution boundedness in the H1-norm, which represents an improvement over the L∞-norm boundedness reported in previous literature. Secondly, relying on this boundedness and the high-precision results of the bilinear element, we derive the unconditional error estimates for superclose and global superconvergence without any restrictions on the time step size Δt and spatial mesh size h. Finally, numerical examples are presented to validate the theoretical analysis.

Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method

We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods are a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.

Effect of high humidity on the initial mechanical properties and failure modes of thin spray-on liners (TSLs)

Thin Spray-on Liners (TSLs) are a novel rock support technology used in underground projects, which often encounter harsh environments such as high humidity. The mechanical properties of TSLs are affected by the high humidity environment. This study aims to summarize the support mechanisms of TSLs and evaluate the effects of humidity on their performance. Two cement-based TSLs were prepared. They used vinyl acetate-ethylene copolymer emulsion (VAE emulsion) and polyacrylate emulsion as modifying materials, respectively. Specimens were subjected to four different humidity environments for 7 days. The mechanical properties of the specimens were evaluated through compressive, flexural and pull-out tests. The impermeability test was utilized to test the sealing of TSLs. Microscopic observation was conducted using a Lecia microscope. The study also analyzed the interfacial failure modes observed in the pull-out test. The results indicated that increasing humidity hindered the film formation reaction of TSLs, preventing the formation of a continuous three-dimensional spatial polymer mesh structure. Instead, cohesive large particles were formed, negatively impacting the mechanical properties. Additionally, increasing humidity led to the formation of large pores and crack lines within TSLs, further decreasing their mechanical and sealing properties. The increase in humidity also weakened the dry shrinkage effect of TSLs, resulting in increased bond strength. Therefore, analyzing the initial mechanical properties of TSLs under high humidity conditions is crucial for guiding their application in underground projects.

TGN: A Temporal Graph Network for Physics Prediction

Long-term prediction of physical systems on irregular unstructured meshes is extremely challenging due to the spatial complexityof meshes and the dynamic changes over time; namely, spatial dependence and temporal dependence. Recently, graph-based next-step prediction models have achieved great success in the task of modeling complex high-dimensional physical systems. However, due to these models ignoring the temporal dependence, they inevitably suffer from the effects of error accumulation. To capture the spatial and temporal dependence simultaneously, we propose a temporal graph network (TGN) to predict the long-term dynamics of complex physical systems. Specifically, we introduce an Encode-Process-Decode architecture to capture spatial dependence and create low-dimensional vector representations of system states. Additionally, a temporal model is introduced to learn the dynamic changes in the low-dimensional vector representations to capture temporal dependence. Our model can capture spatiotemporal correlations within physical systems. On some complex long-term prediction tasks in fluid dynamics, such as airfoil flow and cylinder flow, the prediction error of our method is significantly lower than the competitive GNN baseline. We show accurate phase predictions even for very long prediction sequences.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Quantifying and eliminating the time delay in stabilization exponential time differencing Runge–Kutta schemes for the Allen–Cahn equation

Although the stabilization technique is favorable in designing unconditionally energy stable or maximum-principle-preserving schemes for gradient flow systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen–Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen–Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the mesh-dependent maximum principle and energy stability, we investigate a family of exponential time differencing Runge–Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge–Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes.

طريقة بيتروف جاليركين الخطية المنفصلة لمعادلات الانتشار الكسرية للزمن

We propose and analyze piecwise linear discontinuous Petrov-Galerkin method in time combined with a standard conforming finite element method in space for the numerical solution of time-fractional diffusion problems of order 0 < μ < 1. We prove the stability of the exact solution. The existence, uniqueness and stability of approximate solutions will be proved. We employ a non-uniform mesh based on concentrating the cells near the singularity. The advantage of employing a non-uniform mesh is improving the accuracy of the approximate solution. Numerical experiments indicate the error in L∞(0, T ; L2(Ω))-norm is of order kmin(γ(1- μ),2) + h2, where k denotes the maximum time steps and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh and γ > 0.