Convergence Of Numerical Solution Research Articles

Approximations of the Euler–Maruyama Method of Stochastic Differential Equations with Regime Switching

This work focuses on a class of regime-switching diffusion processes with both continuous components and discrete components. Under suitable conditions, we adopt the Euler–Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we first show the convergence rates in the Lp-norm of the stochastic differential equations with regime switching under Lipschitz conditions. Then, we also discuss L1 and L2 convergence under non-Lipschitz conditions.

Honeycomb-configured dissipative nanofluid flow within a squeezed channel with entropy generation: regression and numerical evaluations

PurposeNanosized honeycomb-configured materials are used in modern technology, thermal science and chemical engineering due to their high ultra thermic relevance. This study aims to scrutinize the heat transmission features of magnetohydrodynamic (MHD) honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts.Design/methodology/approachMass, energy and momentum preservation laws are assumed to find the mathematical model. A set of unified ordinary differential equations with nonlinear behavior is used to express the correlated partial differential equations of the established models, adopting a reasonable similarity adjustment. An approximate convergent numerical solution to these equations is evaluated by the shooting scheme with the Runge–Kutta–Fehlberg (RKF45) technique.FindingsThe impression of pertinent evolving parameters on the temperature, fluid velocity, entropy generation, skin friction coefficients and the heat transference rate is explored. Further, the significance of the irreversibility nature of heat transfer due to evolving flow parameters are evaluated. It is noted that the heat transference rate performance is improved due to the imposition of the allied magnetic field, Joule dissipation, heat absorption, squeezing and thermal buoyancy parameters. The entropy generation upsurges due to rising magnetic field strength while its intensification is declined by enhancing the porosity parameter.Originality/valueThe uniqueness of this research work is the numerical evaluation of MHD honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts. Furthermore, regression models are devised to forecast the correlation between the rate of thermal heat transmission and persistent flow parameters.

Axial Green Function Method in an Efficient Projection Scheme for Incompressible Navier–Stokes Flow in Arbitrary Domains

We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.

Numerical analysis of age-structured HIV model with general transmission mechanism

In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number R0Δt. To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers R0Δt is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if R0Δt<1. Moreover, for R0Δt>1, a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results.

K voprosu o chislennom reshenii nekonservativnykh giperbolicheskikh sistem uravneniy

Issues related to the lack of convergence in the application of formally path-conservative difference schemes for solving nonconservative hyperbolic systems of equations are numerically investigated. This problem is central in constructing well-posed difference schemes for solving this class of problems. The basic concepts of the theory of nonconservative hyperbolic systems of equations and the corresponding problems of constructing difference schemes for their solution are outlined. A variant of the HLL method is proposed that allows using an arbitrary explicitly specified path. For a model system of Burgers equations, the shock adiabates and paths corresponding to the viscous regularization of a system of a given form are explicitly calculated. The reasons for the lack of convergence of numerical solutions of exact ones in the case of incorrect application of the corresponding algorithms are analyzed. It is shown that, at least in the particular case considered, a variant of the HLL method that is formally conservative along the way gives the correct solution of the problem.

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

Variational inequalities of multilayer viscoelastic systems with interlayer Tresca friction: Existence and uniqueness of solution and convergence of numerical solution

The mathematical model of multilayer viscoelastic system based on the actual structure of asphalt pavement and the properties of its numerical solutions are studied. Firstly, based on the partial differential equations, the variational inequality model of the multilayer viscoelastic system is derived, and the existence and uniqueness of its solutions are further proved. Then, based on the finite element theory, the convergence property and error analysis of the semi‐discrete numerical solutions of the variational inequalities are further studied. Finally, the convergence and error analysis of the fully discrete numerical solutions of the variational inequalities are derived by the forward difference method. The conclusion of the above error analysis also confirms the feasibility of studying the mechanical response of asphalt pavement based on variational inequality.

Error estimates of a two‐grid penalty finite element method for the Smagorinsky model

In the paper, we consider the penalty finite element methods (FEMs) for the stationary Smagorinsky model. Firstly, a one‐grid penalty FEM is proposed and analyzed. Since this method is nonlinear, a novel linearized iteration scheme is derived for solving it. We also derived the stability and convergence of numerical solutions for this iteration scheme. Furthermore, a two‐grid penalty FEM is developed for Smagorinsky model. Under , this method consist of solving a nonlinear Smagorinsky model by the one‐grid penalty FEM with the proposed linearized iteration scheme on a coarse mesh with mesh width and then solving a linearized Smagorinsky model based on the Newton iteration on a fine mesh with mesh width , respectively. Stability and error estimates of numerical solutions for two‐grid penalty FEM are presented. Finally, some numerical tests are provided to confirm the theoretical analysis and the effectiveness of the developed methods.

Bi-level algorithm for optimizing hyperparameters in penalized nonnegative matrix factorization

Learning approaches rely on hyperparameters that impact the algorithm’s performance and affect the knowledge extraction process from data. Recently, Nonnegative Matrix Factorization (NMF) has attracted a growing interest as a learning algorithm. This technique captures the latent information embedded in large datasets while preserving feature properties. NMF can be formalized as a penalized optimization task in which tuning the penalty hyperparameters is an open issue. The current literature does not provide any general framework addressing this task. This study proposes to express the penalty hyperparameters problem in NMF in terms of a bi-level optimization. We design a novel algorithm, named Alternating Bi-level (AltBi), which incorporates the hyperparameters tuning procedure into the updates of NMF factors. Results of the existence and convergence of numerical solutions, under appropriate assumptions, are studied, and numerical experiments are provided.

Convergence of numerical solution for the inhomogeneous Landau‐Lifshitz equations with Gilbert damping

In this paper, we discuss the inhomogeneous Landau‐Lifshitz equation with nonuniform Gilbert damping term by numerical method. First, we establish a semi‐discrete form for the inhomogeneous Landau‐Lifshitz equation with nonuniform Gilbert damping, which is continuous in time. Then we study the temporal discretization. It proposes a simple projection method to solve our issue. Finally, we prove that it is unconditionally stable.

Convergence of first‐order finite volume method based on exact Riemann solver for the complete compressible Euler equations

AbstractRecently developed concept of dissipative measure‐valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first‐order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral and the Richtmyer‐Meshkov problem are consistent with our theoretical analysis.

The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies

Finite element analysis of complex bodies is frequently used in design to determine the size of deformations. Successive iterations, with progressive refinement of mesh densities, are most often required to obtain a sufficiently accurate convergent numerical solution. This process is costly, time consuming, and requires superior hardware and software. The paper presents a quick and effortless way to determine a sufficiently accurate value of the numerical solution. The mentioned solution is obtained by amending the numerical solution resulting for a certain value of the mesh density of the studied body with an adequate proportionality coefficient determined following the deformation study of simple bodies differently subject to external forces. It is assumed that the elastic displacement of the various bodies has a similar evolution as the mesh density increases and that the values of the proportionality coefficients considered are approximately equal for identical mesh densities. Examples presented are related to the reference body of the mechanical press PAI 25.

Mixed Convective Flow in Horizontal Rectangular Enclosure with a Wall Heat Source: A Numerical Study

In this work, we have investigated the heat transfer of a mixed convective flow in horizontal rectangular enclosure having a wall heat source. For the purpose of computing stable and convergent numerical solutions of the flow variables, the governing equations along with the boundary conditions are solved by using the upwind finite volume scheme and a modified SIMPLE algorithm. For discussing the heat transfer from the bottom wall of the enclosure, the average Nusselt numbers are computed from relevant empirical correlations existing in the literature. These correlations are in the range of 1708 ≤ Ra ≤ 108 and 1708 ≤ Ra ≤ 3.5×109 in terms of different Rayleigh numbers for the Newtonian fluids such as air and water. Numerical solutions obtained from this study are compared with available benchmark solutions.

A Computationally Fast Method to Simulate Microwave-Heated Monoliths

Microwave-assisted process intensification is a promising technique hindered by the unavailability of design rules for optimization and scale-up. High-throughput computational models allowing the exploration of a vast design space are imperative for the rapid development and deployment of intensified microwave reactors. Recently, structured reactors, especially monoliths, are emerging as a canonical multiphase reactor setup in the microwave-heating community. The vast separation of scales in these reactors leads to a large number of mesh elements, making the simulations time-consuming and challenging to converge. To this end, we employ volume and asymptotic averaging to represent monoliths, a multiphase system consisting of a fluid and a solid phase, as a continuum or effective medium. We rigorously verify the adequacy of the averaging techniques to predict the spatiotemporal distribution of the electric field, electromagnetic power dissipation, and temperature against fully resolved monolith simulations. The developed continuum model can replicate the three-dimensional transient behavior obtained from the multiphase simulations with an order of magnitude lower computational expense. Moreover, the continuum model allows easier mesh generation and convergence of numerical solution than the multiphase model. The developed multiscale framework can be used to simulate microwave-heated monoliths and other multiphase reactors, such as packed beds and open-cell foams.

Chute flows of dry granular media: Numerical simulations by a well-posed multilayer model and comparisons with experiments

Debris flows and avalanches are dangerous natural phenomena, characterized by the gravity-driven motion of granular media immersed in a fluid. For an appropriate hazard assessment or disaster mitigation by scenario investigation, it is crucial to capture the underlying dynamics of the granular solid phase. For this purpose, a multilayer depth-averaged approach represents a promising and computationally efficient tool over fully three-dimensional models. Here we use a mathematically well-posed multilayer model, which implements the µ(I)-rheology and a dilatancy law depending on the inertial number, I, and compare the numerical results of the model with laboratory experiments of steady uniform chute flows over an erodible bed. The well-posedness of the model for any value of I, which is essential to get convergent numerical solutions, is achieved by considering an approximation of the in-plane stress gradients, directly emerging from the µ(I)-rheology. The predicted velocity profiles show a very good agreement with the experimental ones, measured by particle image velocimetry (PIV). The volume fraction profiles by the multilayer model are also in good qualitative agreement with those measured by using the stochastic-optical method (SOM), while they tend to overestimate the volume fraction measurements in the more dilute upper region, closer to the free surface.

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Static aeroelastic instability analysis of an inverted plate in ground effect and comparison with a wind-tunnel test

This paper visits the static instability of an inverted cantilevered plate in axial flow with the ground effect. This plate model is clamped at the trailing edge, and its leading-edge is allowed to move freely. A theoretical analysis strategy and corresponding convergent numerical solutions are reported and compared with a wind-tunnel experiment. In theory, we try a new way to study such an instability problem. We derive the instability equation by operator theory in state-space formulation and model such an instability problem as a mathematical function approximation problem. Glauert’s expansion and the least square method are applied for the numerical solutions of vorticity in ground effect using the mirror image method. The unknown function of the plate slope is expanded as a series of essential polynomial functions based on the Weierstrass theorem, and the least square method is applied for its numerical solutions. The experimental study develops a multi-point contact method using a control rod to preload the tested plate with a slight static deformation close to its natural modal shape. The contact state change between the initially deformed tested plate and the control rod is reflected by the evolution of the voltage output of a designed circuit. A criterion for determining the critical condition of plate instability is established based on such voltage change, and it has a better anti-interference ability to the flow. Results show that the present theoretical analysis agrees well with the experiment and other archived theories. The critical plate slopes (i.e., the plate instability modes) are not dependent on the ground confinement. This conclusion allows us to establish an analytical approximation between the critical dynamic pressure and the ground confinement. The present study and results can successfully predict the plate instability in the ground effect.

Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality

This paper investigates a symmetric dual-wind discontinuous Galerkin (DG) method for solving parabolic variational inequalities. By employing a symmetric dual-wind DG discretization in space and a backward Euler discretization in time, we propose a fully discrete scheme to solve a time-dependent obstacle problem. Under reasonable regularity assumptions on the exact solution, we prove the convergence of numerical solutions with rates in the L∞(L2) and L2(H1)-like energy errors by introducing a new interpolation operator which is a combination of the standard interpolation operator and a positive-preserving interpolation operator. Numerical experiments are provided to validate the effectiveness of the proposed method.

A weak Galerkin finite element method for the Kelvin-Voigt viscoelastic fluid flow model

In this paper, we consider a weak Galerkin finite element method for the Kelvin-Voigt viscoelastic fluid flow model. Firstly, the weak Galerkin finite element method is used to approximate the spatial variable and we use piecewise polynomials of degrees k, k−1 and k−1(k≥1) to approximate the velocity, pressure, and the numerical trace of the velocity on the interfaces of elements, respectively. Secondly, the backward Euler difference method is adopted in the temporal discretization for the fully discrete scheme. Furthermore, the stability and optimal convergence of numerical solutions in L∞(L2) and L∞(H1)-norms of velocity as well as L∞(L2)-norm of pressure were presented. Finally, numerical examples verify the effectiveness of the proposed method, which also obtain that the algorithm has convergence and robust for different retardation time.

Comparison Convergent Numerical Result for Fractal Caputo and Fractal Caputo Fabrizo

The new numerical technicality operational matrix based on shifted Legendre polynomials to solve the linear and nonlinear meaning of the fractal fractional of Caputo and Caputo-Fabrizio. By using this method, we get perfect and precise results. Studying the comparative of the convergence of numerical solutions with the accuracy of the fractal fractional Caputo and the fractal fractional Caputo-Fabrizio .