Forward Euler Scheme Research Articles

On different implementations of boundary conditions in the meshless RBF-FD method for phase-field modelling of dendritic solidification

Dendritic morphology is one of the most common microstructures in solidifying metallic materials. The phase-field method represents one of the most suitable approaches for modelling the evolution of dendritic morphology. In this paper, the meshless RBF-FD method and forward Euler scheme are used to solve the partial differential equations arising from the phase-field model for dendritic growth. We consider the growth of a single dendrite into a supercooled pure melt. On the computational domain’s surfaces, we apply symmetric boundary conditions. Special care has to be taken in the RBF-FD method to satisfy boundary conditions accurately. In this paper, we test four different implementations of boundary conditions. In the first one, the boundary conditions are incorporated when constructing an interpolation problem in local support domains containing boundary nodes. In the second one, the interpolation problem in the local support domains does not consider boundary conditions, which are satisfied by solving a system of linear equations for values in all boundary nodes at each time step. The third one complements the second one with the use of ghost nodes. The fourth implementation is an alternative one, where the values in ghost nodes are determined by direct mirroring. The accuracy and computational efficiency of all four implementations are compared. We discuss the advantages and disadvantages of each implementation. We show that using ghost nodes is recommended for implementing Neumann boundary conditions in the RBF-FD method.

Fast explicit time integration schemes for parabolic problems in mechanics

AbstractWe present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

The impact of standard and nonstandard finite difference schemes on HIV nonlinear dynamical model

Mathematical models are enormously valuable in recognition the characteristics of infectious afflictions. The present study describes and analyses a nonlinear Susceptible-Infected (S·I) type mathematical model for HIV/AIDS. To better comprehend the dynamics of disease diffusion, it is assumed that by giving AIDS patients timely Anti Retroviral Therapy (ART), their transition into HIV infected class is attainable. The ART treatment can reduce or manage the spread of disease among individuals that can extend their life for some more years. For the model, the basic reproduction number is formed which provides a base to study the stability of disease free and endemic equilibria. To understand the entire dynamical behavior of the model, standard finite difference (SFD) schemes such as Runge-Kutta of order four (RK-4) and forward Euler schemes and nonstandard finite difference (NSFD) scheme are implemented. The goal of constructing the NSFD scheme for differential equations is to ensure that it is dynamically reliable, while maintaining important dynamical properties like the positivity of the solutions and its convergence to equilibria of continuous model for all finite step sizes. However, the essential characteristics of the continuous model cannot be properly maintained by the Euler and RK-4 schemes, leading to the possibility of numerical solutions that are not entirely similar to those of the original model. For the NSFD scheme, the Routh-Hurwitz criterion is used to assess the local stability of disease-free and endemic equilibria. To explain the global stability of both the equilibria, Lyapunov functions are offered. To verify the theoretical findings and validate the dynamical aspects of the abovementioned schemes, numerical simulations are also provided. The outcomes offered in this study may be engaged as an effective tool for forecasting the progression of HIV/AIDS epidemic diseases.

Inertial instability and phase error in Euler forward predictor-corrector time integration schemes: Improvement of modeling Great Lakes thermal structure and circulation using FVCOM

This study investigates the inertial stability properties and phase error of numerical time integration schemes in several widely-used ocean and atmospheric models. These schemes include the most widely used centered differencing (i.e., leapfrog scheme or the 3-time step scheme at n-1, n, n+1) and 2-time step (n, n+1) 1st-order Euler forward schemes, as well as 2nd-stage and 3rd- and 4th-stage Euler predictor-corrector (PC) schemes. Previous work has proved that the leapfrog scheme is neutrally stable with respect to the Coriolis force, with perfect inertial motion preservation, an amplification factor (AF) equal to unity, and a minor overestimation of the phase speed. The 1st-order Euler forward scheme, on the other hand, is known to be unconditionally inertially unstable since its AF is always greater than unity. In this study, it is shown that 3rd- and 4th-order predictor-corrector schemes 1) are inertially stable with weak damping if the Coriolis terms are equally split to n+1 (new value) and n (old value); and 2) introduce an artificial computational mode. The inevitable phase error associated with the Coriolis parameter is analyzed in depth for all numerical schemes. Some schemes (leapfrog and 2nd-stage PC schemes) overestimate the phase speed, while the others (1st-order Euler forward, 3rd- and 4th-stage PC schemes) underestimate it. To preserve phase speed as best as possible in a numerical model, alternating a scheme that overestimates the phase speed with a scheme that underestimates the phase speed is recommended. Considering all properties investigated, the leapfrog scheme is still highly recommended for a time integration scheme. As an example, a comparison between a leapfrog scheme and a 1st-order Euler forward scheme is presented to show that the leapfrog scheme reproduces much better vertical thermal stratification and circulation in the weakly-stratified Great Lakes.

A coupled domain–boundary type meshless method for phase-field modelling of dendritic solidification with the fluid flow

PurposeThis study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method.Design/methodology/approachA preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case.FindingsThe solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented.Originality/valueA combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.

Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counterpart of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior calculus operators. We prove the boundedness of the method for the forward Euler and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme to handle large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

Application of a meshless space-time adaptive approach to phase-field modelling of polycrystalline solidification

We have developed a 2-D numerical meshless adaptive approach for phase-field modelling of dendritic solidification. The quadtree-based approach decomposes the computational domain into quadtree sub-domains of different sizes. The algorithm generates uniformly-distributed computational nodes in each quadtree sub-domain. We apply the meshless radial basis function generated finite difference method and the forward Euler scheme to discretise governing equations in each computational node. The fixed ratio between the characteristic size and the node spacing of a quadtree sub-domain ensures space adaptivity. The adaptive time-stepping accelerates the calculations further. In the framework of previous research studies, we used the approach to solve quantitative phase-field models for single dendrite growth in pure melts and dilute binary alloys. In the present study, we upgrade the solution procedure for the modelling growth of multiple differently oriented dendrites. Along with the space-time adaptive approach, we apply non-linear preconditioning of the phase-field equation to increase computational efficiency. We investigate a novel numerical approach’s accuracy and computational efficiency by simulating the equiaxed dendrite growth from a dilute binary alloy.

Stability of non-linear flow in heterogeneous porous media simulations using higher order and continuity basis functions

Using isogeometric analysis, we present a highly optimized FORTRAN code for simulations of three-dimensional non-linear flow problem in heterogeneous media. This includes deriving the weak formulations, discretizing with the forward Euler scheme, and implementing computation of the right-hand side. Some properties of exact solutions of heat transport and non-linear flow in heterogeneous media problems have been derived to validate the numerical solution. Stability analysis of the employed explicit time-stepping scheme has been performed. First, the properties of the discrete form of the heat transfer problem have been systematically investigated, leading to the conjecture concerning the sufficient condition of stability of the scheme, which numerical computations have subsequently confirmed. A similar analysis has been carried out for the non-linear flow in heterogeneous media problem. For this problem, the bound on the time step was conjectured to decrease exponentially, which has been verified experimentally. The simulation’s behavior close to the stability limit has been investigated and found to be considerably more complex than for the heat transport problem. Detailed experimental analysis of the time complexity of the particular parts of the simulator algorithm has been described. In addition, dependence on both mesh size and order of continuity of B-spline basis has been investigated.

Modified unified critical state model for soils considering over-consolidation and cyclic loading behaviours

This paper presents a modified unified critical state model to predict the mechanical responses of both clays and sands under over-consolidation and cyclic loading conditions on the basis of clay and sand model (CASM), which is named as CASM-kII. Through the application of subloading surface concept, CASM-kII is able to describe the plastic deformation inside the yield surface and the reverse plastic flow, and is thus expected to capture the over-consolidation and cyclic loading behaviours of soils. CASM-kII is numerical implemented by the using of the forward Euler scheme with automatic substepping and error control. Then, a sensitivity study is carried out to check the influences of the three new parameters of CASM-kII on the mechanical response of soils in over-consolidation and cyclic loading conditions. Through the comparisons of experimental data and simulated results, it is found that CASM-kII is able to satisfactorily describe the mechanical responses of both clays and sands in over-consolidation and cyclic loading conditions.

Spatio-temporal numerical modeling of stochastic predator-prey model

In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder’s fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model.

Numerical‐discrete‐scheme‐incorporated recurrent neural network for tasks in natural language processing

AbstractA variety of neural networks have been presented to deal with issues in deep learning in the last decades. Despite the prominent success achieved by the neural network, it still lacks theoretical guidance to design an efficient neural network model, and verifying the performance of a model needs excessive resources. Previous research studies have demonstrated that many existing models can be regarded as different numerical discretizations of differential equations. This connection sheds light on designing an effective recurrent neural network (RNN) by resorting to numerical analysis. Simple RNN is regarded as a discretisation of the forward Euler scheme. Considering the limited solution accuracy of the forward Euler methods, a Taylor‐type discrete scheme is presented with lower truncation error and a Taylor‐type RNN (T‐RNN) is designed with its guidance. Extensive experiments are conducted to evaluate its performance on statistical language models and emotion analysis tasks. The noticeable gains obtained by T‐RNN present its superiority and the feasibility of designing the neural network model using numerical methods.

BIFURCATION AND COMPARISON OF A DISCRETE-TIME HINDMARSH-ROSE MODEL

In this paper, a Hindmarsh-Rose model discretized by a nonstandard finite difference (NSFD) scheme is considered. Bifurcation behaviors are compared between the model obtained by the forward Euler scheme and the model obtained by the NSFD scheme. Through analytical and numerical comparisons, the Neimark-Sacker bifurcation of the model discretized by the NSFD method is closer to the Hopf bifurcation of the original continuous Hindmarsh-Rose model than that discretized by the forward Euler method. Moreover, due to the NSFD method's better stability and convergence, the integral step size can be chosen larger in the NSFD scheme. And much more dynamic behaviors can be obtained by using the NSFD scheme than those in the forward Euler scheme. These confirmed results can at least guarantee true available numerical results to investigate complex neuron dynamical systems.

The Existence of a Limit-Cycle of a Discrete-Time Lotka-Volterra Model with Fear Effect and Linear Harvesting

Modeling the interaction between prey and predator plays an important role in maintaining the balance of the ecological system. In this paper, a discrete-time mathematical model is constructed via a forward Euler scheme, and then studied the dynamics of the model analytically and numerically. The analytical results show that the model has two fixed points, namely the origin and the interior points. The possible dynamical behaviors are shown analytically and demonstrated numerically using some phase portraits. We show numerically that the model has limit-cycles on its interior. This guarantees that there exists a condition where both prey and predator maintain their existence periodically.

On the fractional-order glucose-insulin interaction

<abstract><p>We consider a fractional-order model of glucose and insulin interaction based on the intra-venous glucose tolerance test (IVGTT). We show the existence of the model's solution, uniqueness, non-negativity, and boundadness. In addition, for the proposed fractional-order model, we establish sufficient conditions for stability or instability. Some conditions for bifurcation in the proposed model are presented using bifurcation theory. Further, in the case of first order the model is discretized by applying the forward Euler scheme. We investigate how small the time step size must be chosen to guarantee that the steady state solution is an attractive fixed point of the discretized model. Numerical simulations that we provided support the analytical results.</p></abstract>

Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross–Pitaevskii equation in dispersive media

In this study, the Gross–Pitaevskii equation perturbed with multiplicative time noise is under consideration numerically and analytically. The NLSE is a universal governing model that helps in evolution of complex fields that are used in dispersive media. For the numerical solution, we used the stochastic forward Euler (SFE) scheme. To find the exact solutions, we chose the techniques namely [Formula: see text]-model expansion. For the analysis of the proposed scheme, we checked the stability of the scheme with the help of Von-Neumann criteria and the consistency of the scheme with the mean of Ito’s sense. The exact solutions of the model are constructed successfully in the Jacobi elliptic function in the form of trigonometric and hyperbolic functions. Last, we compared the graphical behavior of the proposed scheme with some exact solutions by using the unique selection of initial and boundary conditions. The plots are constructed in the form of 3D, line, and contour representation by choosing the different values of parameters.

Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross–Pitaevskii equation in dispersive media

In this study, the Gross–Pitaevskii equation perturbed with multiplicative time noise is under consideration numerically and analytically. The NLSE is a universal governing model that helps in evolution of complex fields that are used in dispersive media. For the numerical solution, we used the stochastic forward Euler (SFE) scheme. To find the exact solutions, we chose the techniques namely [Formula: see text]-model expansion. For the analysis of the proposed scheme, we checked the stability of the scheme with the help of Von-Neumann criteria and the consistency of the scheme with the mean of Ito’s sense. The exact solutions of the model are constructed successfully in the Jacobi elliptic function in the form of trigonometric and hyperbolic functions. Last, we compared the graphical behavior of the proposed scheme with some exact solutions by using the unique selection of initial and boundary conditions. The plots are constructed in the form of 3D, line, and contour representation by choosing the different values of parameters.

Acceleration of RBF-FD meshless phase-field modelling of dendritic solidification by space-time adaptive approach

• A novel meshless approach for phase-field modelling of dendritic growth. • Reduction of discretisation-induced anisotropy by the scattered nodes. • Increase in computational efficiency by a space-time adaptive approach. A novel adaptive numerical approach is developed for an accurate and computationally efficient phase-field modelling of dendritic solidification. The adaptivity is based on the dynamic quadtree domain decomposition. A quadtree decomposes the computational domain into rectangular sub-domains of different sizes. Each sub-domain is extended to ensure overlap communication between neighbouring sub-domains. In each sub-domain, uniform distribution of computational nodes is generated. The product between the node density and the sub-domain area is fixed to ensure the h-adaptivity. The adaptive approach provides the highest density of computational nodes at the solid-liquid interface and the lowest density in the bulk of the phases. The meshless radial basis function generated finite difference (RBF-FD) method is applied for the spatial discretisation of the partial differential equations which arise from the phase-field model. The RBF-FD method is especially appealing since it allows straightforward spatial discretisation of partial differential equations on scattered node distributions. The use of scattered node distribution reduces the discretisation-induced anisotropy in the phase-field modelling of dendritic growth. The forward Euler scheme is used for temporal discretisation. The adaptive time-stepping is employed to speed up the calculations further. The performance of the novel numerical approach is tested for dendritic solidification of supercooled pure melts and supersaturated dilute binary alloys at arbitrary preferential growth directions. The impact of the numerical parameters on the accuracy and computational efficiency is thoroughly analysed. It is shown that the RBF-FD method, defined on scattered node distribution, together with the space-time adaptive approach, represents an accurate and efficient technique for solving the phase-field models of dendritic solidification.

Positivity preserving numerical method for epidemic model of hepatitis B disease dynamic with delay factor

This work attempts to study the numerical solution of nonlinear delayed Immunized Susceptible Latent Infected and Recovered (MSLIR) epidemic model of HBV disease. Reproduction number, equilibria and stability are discussed. Three different numerical techniques, Euler, RK-4 and the non-standard finite difference (NSFD) techniques are used for the numerical solution of the model. The proposed technique is independent of the size of the time step, while forward Euler and RK-4 depend on the size of a time step and retains all essential characteristics of the continuous MSLIR epidemic model like positivity and stability of equilibrium, while well-known forward Euler and RK-4 cannot sustain these characteristics. Therefore, the proposed (NSFD) technique becomes a more efficient and reliable numerical technique than the forward Euler and RK-4 scheme. Numerical simulations are presented for the validation of the obtained results.

Multistability, chaos and mean population density in a discrete-time predator–prey system

We investigate a discrete-time system derived from the continuous-time Rosenzweig–MacArthur (RM) model using the forward Euler scheme with unit integral step size. First, we analyze the system by varying carrying capacity of the prey species. The system undergoes a Neimark–Sacker bifurcation leading to complex behaviors including quasiperiodicity, periodic windows, period-bubbling, and chaos. We use bifurcation theory along with numerical examples to show the existence of Neimark–Sacker bifurcation in the system. The transversality condition for the Neimark–Sacker bifurcation at the bifurcation point is derived using a different approach compared to the existing ones. Multistability of different kinds: periodic–periodic and periodic–chaotic are also revealed. The basins of attraction for these multistabilities show complicated structures. The sufficient increase in the nutrient supply to the prey species may have negative effect in form of decrease in mean predator stock which leads to extinction of predator. Therefore, the paradox of enrichment is prominent in our discrete-time system. Further, we introduce prey and predator harvesting to the system. When the system is subjected to prey (or predator) harvesting, it stabilizes into equilibrium state. The system also exhibits complicated dynamics including multistability and Neimark–Sacker bifurcation when prey (or predator) harvesting rate is varied. With prey harvesting, the mean predator density increases when the system exhibits nonequilibrium dynamics. However, we have identified a situation for which the unstable equilibrium predator biomass decreases while mean predator density increases under predator mortality. Thus, this counter-intuitive phenomenon (positive effect on predator biomass) referred to as hydra effect is detected in our discrete-time system.

A dynamically consistent approximation for an epidemic model with fuzzy parameters

In this article, a susceptible, infectious, and recovered (SIR) model with fuzzy parameters is discussed. These concepts are uncertain due to the different degrees of susceptibility, infectivity, and recovery among the individuals in the population. Differences may appear when the populations considered have different ages, habits, customs, and different levels of resistance, etc. More realistic models are needed which consider these different degrees of susceptibility, infectivity, and recovery of the individuals to overcome the above uncertainties. The existence and uniqueness of the model are studied. Forward Euler and NSFD methods were employed to solve the model in classical and fuzzy senses. The developed fuzzy NSFD method preserves positivity and convergence which are the main features of a dynamic system. To the best of our knowledge, the numerical analysis of the studied SIR model using the forward Euler, fuzzy Euler, NSFD, and FNSFD schemes has not been specifically studied in the literature.