Discretization Scheme Research Articles

Convergence analysis of a decoupled pressure-correction SAV-FEM for the Cahn–Hilliard–Navier–Stokes model

In this paper, a linear, decoupled and unconditional energy-stable numerical scheme of Cahn–Hilliard–Navier–Stokes (CHNS) model is constructed and analyzed. We reformulate the CHNS model into an equivalent system based on the scalar auxiliary variable approach. The Euler implicit/explicit scheme is used for time discretization, that is, linear terms are implicitly processed, nonlinear terms are explicitly processed, and pressure-correction method of the Navier–Stokes equations is also adopted. In this way, we achieve the decoupling of phase field, velocity and pressure, and only need to solve a series of linear equations with constant coefficients at each time step, thereby improving computational efficiency. Then the finite element method is used for space discretization, and the finite element spaces of phase field, velocity and pressure are taken as Pl−Pl−Pl−1 respectively. We also verify that the proposed scheme satisfies the discrete energy dissipation law. The optimal error estimates of the fully discrete scheme is proved, which the phase field, velocity and pressure satisfy the first-order accuracy in time and the (ℓ+1,ℓ+1,ℓ)th order accuracy in space, respectively. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A novel α-absolute value preconditioner for all-at-once systems from heat equations

In this paper, we generalize a fast Fourier transforms (FFTs) based preconditioner and propose a novel α-absolute value preconditioner for all-at-once systems from heat equations. Our preconditioner is symmetric positive definite and can also be computed efficiently using fast Fourier transforms or discrete sine transforms (DSTs). Furthermore, we prove that the corresponding preconditioned matrix has a unitary-plus-low-rank decomposition, and the eigenvalues of the preconditioned matrix are clustered at ±1, which leads to fast convergence rate of the preconditioned MINRES method. Finally, numerical examples on all-at-once systems from heat equations with different discretization scheme demonstrate the effectiveness of our new preconditioner.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

Discretization of prescribed-time observers in the presence of noises and perturbations

An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Stiffness of reinforced concrete structures under bending considering shear and axial forces (part 2)

The paper presents a physical and computational model for determining the parameters of limit states of reinforced concrete structures under complex stressed state - bending with axial and shear forces. Based on the adopted scheme of cross-section discretization and A.R. Rzhanitsyn's theorem of duality between force and deformation parameters, the forward and backward transfer for determining the coefficients of the stiffness matrix of reinforced concrete rod structures with inclined and normal cracks have been constructed. Stiffness of structures in the zone of inclined cracks is determined using the model of composite strips which separate the zone with inclined cracks. The hypothesis about the character of deformation distribution in a complex-stressed reinforced concrete element with inclined cracks is accepted.For this model the conditional shear modulus is obtained, which allows to determine the average relative linear and angular strains of concrete and reinforcement at the point adjacent to the shear joint between inclined cracks. Based on this model and using the experimentally obtained value of the relative shear in the inclined crack, the dowel forces in the reinforcing bar intersected by the inclined crack were determined. The application of the obtained analytical relationships in the practice of designing reinforced concrete structures allows not only to clarify significantly the definition of displacements and width of opening of inclined and normal cracks, but also to maximize the convergence of the design and physical model based on experimental data.

Energy dissipation and evolutions of the nonlocal Cahn-Hilliard model and space fractional variants using efficient variable-step BDF2 method

In this work, an energy stable BDF2 scheme with general nonuniform time steps is developed for the nonlocal Cahn-Hilliard model to capture the multi-scale behavior of evolution efficiently and accurately. The fully discrete numerical scheme is demonstrated to inherit main physical properties of the continuous model, namely, mass conservation and energy dissipation. Recently, various variants of the Cahn-Hilliard model with nonlocal operators and fractional Laplacian have attracted great attention in terms of theoretical analysis and numerical approximation, while there is less emphasis on the comparisons among the different variant models. By comparing the impact of various parameters and coarsening behaviors, numerical results suggest that the nonlocal Cahn-Hilliard model bears a strong resemblance to the space fractional Cahn-Hilliard model derived from H−1 gradient flow of the nonlocal energy. Nevertheless, the order of the fractional Laplacian appears to affect only the rate of energy decay, without altering the profile of the solution for the fractional model relating to the H−β (0<β<1) gradient flow of the classical Ginzburg-Landau energy functional in one dimensional case.

Mixed convection characteristics in a long horizontal lid‐driven channel with periodically distributed local flow modulators

AbstractThis study explores the effectiveness of periodically placed rotating blades in enhancing heat transfer in a channel. The channel consists of a cold top plate moving at a constant speed and a fixed hot plate at the bottom. Thin rotating blades are placed periodically along the channel's centerline, with the spacing between their axes equal to the channel's height. This paper analyzes a transient, two‐dimensional, laminar flow problem using energy, momentum, and continuity equations. To address the challenges posed by moving blades, the Galerkin finite element method is implemented within an arbitrary Lagrangian–Eulerian framework, employing a triangular mesh discretization scheme. This study comprehensively explores thermal and hydrodynamic characteristics, including overall heat transfer, thermal frequency, and power consumption of the rotating blade for heat transfer in mixed convection scenarios with Richardson numbers (Ri) ranging from 0.1 to 10 at varying rotational frequency of the blade. Outcomes demonstrate that the inclusion of a rotating blade increases heat transfer up to 50% at lower Ri, after which the impact of the rotating blade diminishes and heat transfer reduces up to 20% at higher Ri. In addition, heat transfer enhances with increasing blade frequency up to Ri = 6.5, beyond which the effect of the frequency overturns. Examining thermal and hydrodynamic characteristics reveals that the blade achieves optimal performance when operating at f = 1 and Ri = 3. The study's insights into mixed convection heat transfer offer versatile applications, benefiting industries and equipment such as electronic cooling, chemical reactors, food processing, material fabrication, solar collectors, and nuclear reactor systems. Moreover, the findings are instrumental in the thermal ventilation of buildings and the development of micro‐electromechanical systems.

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders.

Mixed strategy approach destabilizes cooperation in finite populations with clustering coefficient.

Evolutionary game theory, encompassing discrete, continuous, and mixed strategies, is pivotal for understanding cooperation dynamics. Discrete strategies involve deterministic actions with a fixed probability of one, whereas continuous strategies employ intermediate probabilities to convey the extent of cooperation and emphasize expected payoffs. Mixed strategies, though akin to continuous ones, calculate immediate payoffs based on the action chosen at a given moment within intermediate probabilities. Although previous research has highlighted the distinct impacts of these strategic approaches on fostering cooperation, the reasons behind the differing levels of cooperation among these approaches have remained somewhat unclear. This study explores how these strategic approaches influence cooperation in the context of the prisoner's dilemma game, particularly in networked populations with varying clustering coefficients. Our research goes beyond existing studies by revealing that the differences in cooperation levels between these strategic approaches are not confined to finite populations; they also depend on the clustering coefficients of these populations. In populations with nonzero clustering coefficients, we observed varying degrees of stable cooperation for each strategic approach across multiple simulations, with mixed strategies showing the most variability, followed by continuous and discrete strategies. However, this variability in cooperation evolution decreased in populations with a clustering coefficient of zero, narrowing the differences in cooperation levels among the strategies. These findings suggest that in more realistic settings, the robustness of cooperation systems may be compromised, as the evolution of cooperation through mixed and continuous strategies introduces a degree of unpredictability.

Physically based heat exchanger sizing method for the thermal management system of all-electric regional aircraft

Fully electric propulsion systems integrating hydrogen-powered fuel cells and batteries are promising options to reduce the overall climate impact of regional aircraft. However, the increase in low-temperature heat sources aboard the aircraft calls for advanced thermal management system solutions. To address this challenge, this study presents a sizing methodology for ram air heat exchangers in the nacelle-integrated cooling loop of an all-electric regional aircraft based on the ATR-72 platform. Different discretization schemes are compared to identify an optimal sizing method. The results highlight the simplicity and efficiency of the 0D ε-NTU model. Geometric design variables are optimized with respect to drag and mass during a hot-day take-off. The resulting Pareto front reveals a tendency for low airflow outlet temperatures and large diffuser area ratios to result in lightweight designs but in turn, induce high drag and require a large installation space. Comparative analyses of specific optimal ram air duct designs and equivalent skin heat exchangers demonstrate the potential of a second heat sink over a flight mission. The limited heat transfer area of the skin heat exchanger proves insufficient for hot-day take-off and climb but offers advantages during cruise and descent thanks to the reduced drag.

Jump–diffusion productivity models in equilibrium problems with heterogeneous agents

In this paper we adopt a rational expectations framework to formulate general equilibrium models with heterogeneous agents. The productivity dynamics are characterized by a jump–diffusion model, thus allowing to account for sudden and impactful events. The modelling approach utilizes Hamilton–Jacobi–Bellman (HJB) formulations to represent the endogenous decision-making of firms to remain or exit the industry. When firms decide to exit, they are instantaneously replaced by new entrants. This dynamic leads to the development of a probability density function for firms, which satisfies a Kolmogorov–Fokker–Planck (KFP) equation with a source term. Both HJB and KFP formulations involve partial-integro differential operators due to the presence of jumps. Equilibrium models are completed with the household problem and feasibility conditions. Since (semi-)analytical solutions are not available, a numerical methodology is considered. This approach involves a Crank–Nicolson scheme for the time discretization, an augmented Lagrangian active set method and a finite difference discretization for the HJB formulation, and an appropriate finite difference method for the KFP problem. Moreover, Adams–Bashforth schemes are employed to handle integral terms explicitly. For the global equilibrium problem, we introduce a Steffensen algorithm. Numerical examples are provided to showcase the performance of our proposed numerical methodologies and to illustrate the expected behaviour of computed economic variables.

Mixed virtual element methods for elliptic optimal control problems with boundary observations in L2(Γ)

In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings.

On the convergence of discrete dynamic unbalanced transport models

A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, 𝕊n+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou–Brenier, which extends the Kantorovich–Bures and the Wasserstein–Fisher–Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein–Fisher–Rao distance, thanks to the static formulation, we show that such an assumption can be removed.

A fully discrete scheme on piecewise-equidistant mesh for singularly perturbed delay integro-differential equations

. This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems.

Fully conservative difference schemes for the rotation-two-component Camassa–Holm system with smooth/nonsmooth initial data

This paper derives a semi-discrete conservative difference scheme for the rotation-two-component Camassa–Holm system based on its Hamiltonian invariants. Mass, momentum and energy are preserved for the semi-discrete scheme. Furthermore, a fully discrete finite difference scheme is proposed without destroying any one of the conservative laws. Combining a nonlinear iteration with a threshold strategy, the accuracy of the scheme is guaranteed. Meanwhile, this scheme captures the formation and propagation of solitary wave solutions in long time behavior under smooth/nonsmooth initial data. Remarkably, a new type of asymmetric wave breaking phenomenon is revealed in the case of the nonzero rotational parameter.

A Novel Fully Decoupled Scheme for the MHD System with Variable Density

Abstract In this paper, we first establish a novel first-order, fully decoupled, unconditionally stable time discretization scheme for the MHD system with variable density. This scheme successfully decouples all the coupling terms by combining the gauge-Uzawa method and the scalar auxiliary variable (SAV) method. And we prove its unconditional energy stability. Then we give the first-order finite element scheme and its implementation. Furthermore, we perform a rigorous error analysis of the proposed numerical scheme. Finally, we perform some numerical experiments to demonstrate the effectiveness of the decoupling scheme.

Optimal analysis of finite element methods for the stochastic Stokes equations

Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) ) L^\infty (0, T; L^2(\Omega ; L^2)) norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of O ( τ 1 / 2 + h 2 ) O(\tau ^{1/2}+ h^2) in the L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) ) L^\infty (0, T; L^2(\Omega ; L^2)) norm for approximating the velocity, and strong convergence of O ( τ 1 / 2 + h ) O(\tau ^{1/2}+ h) in the L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) ) L^{\infty }(0, T;L^2(\Omega ;L^2)) norm for approximating the time integral of pressure, where τ \tau and h h denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.