Nonstandard Finite Differences Research Articles

The discrete canonical commutation relationship

We use non-standard finite differences to propose a quantum momentum operator to be used when the spectrum of the operator is discrete. The defined discrete operator complies with the discrete versions of the properties that the continuous variable operator has. The discrete derivative is exact for its eigenfunction, that is, exponential functions. We obtain the discrete adjoint of the momentum operator. The canonical commutation relationship between conjugate operators for discrete variables is diagonal along a particular direction.

Numerical solution of differential – difference equations having an interior layer using nonstandard finite differences

This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.

PINN training using biobjective optimization: The trade-off between data loss and residual loss

Physics informed neural networks (PINNs) have proven to be an efficient tool to represent problems for which measured data are available and for which the dynamics in the data are expected to follow some physical laws. In this paper, we suggest a multiobjective perspective on the training of PINNs by treating the data loss and the residual loss as two individual objective functions in a truly biobjective optimization approach.As a showcase example, we consider COVID-19 predictions in Germany and built an extended susceptibles-infected-recovered (SIR) model with additionally considered leaky-vaccinated and hospitalized populations (SVIHR model) to model the transition rates and to predict future infections. SIR-type models are expressed by systems of ordinary differential equations (ODEs). We investigate the suitability of the generated PINN for COVID-19 predictions and compare the resulting predicted curves with those obtained by applying the method of non-standard finite differences to the system of ODEs and initial data.The approach is applicable to various systems of ODEs that define dynamical regimes. Those regimes do not need to be SIR-type models, and the corresponding underlying data sets do not have to be associated with COVID-19.

Nonstandard finite differences numerical methods for a vegetation reaction–diffusion model

In this work we derive NonStandard Finite Differences (NSFDs) (Anguelov and Lubuma, 2001; Mickens, 2020) numerical schemes to solve a model consisting of reaction–diffusion Partial Differential Equations (PDEs) that describes the coexistence of plant species in arid environments (Eigentler and Sherratt, 2019). The new methods are constructed by exploiting a-priori known properties of the exact solution, such as positivity and oscillating behavior in space. Furthermore, we extend the definition of NSFDs inspired by the Time-Accurate and High-Stable Explicit (TASE) (Bassenne et al., 2021) methodology, also exploring the existing connections between nonstandard methods and the Exponential-Fitting (EF) (Ixaru, 1997; Ixaru and Berghe, 2010) technique.Several numerical tests are performed to highlight the best properties of the new NSFDs methods compared to the related standard ones. In fact, at the same cost, the former are much more stable than the latter, and unlike them preserve all the most important features of the model.

A physics-informed neural network to model COVID-19 infection and hospitalization scenarios

In this paper, we replace the standard numerical approach of estimating parameters in a mathematical model using numerical solvers for differential equations with a physics-informed neural network (PINN). This neural network requires a sequence of time instances as direct input of the network and the numbers of susceptibles, vaccinated, infected, hospitalized, and recovered individuals per time instance to learn certain parameters of the underlying model, which are used for the loss calculations.The established model is an extended susceptible-infected-recovered (SIR) model in which the transitions between disease-related population groups, called compartments, and the physical laws of epidemic transmission dynamics are expressed by a system of ordinary differential equations (ODEs). The system of ODEs and its time derivative are included in the residual loss function of the PINN in addition to the data error between the current network output and the time series data of the compartment sizes. Further, we illustrate how this PINN approach can also be used for differential equation-based models such as the proposed extended SIR model, called SVIHR model.In a validation process, we compare the performance of the PINN with results obtained with the numerical technique of non-standard finite differences (NSFD) in generating future COVID-19 scenarios based on the parameters identified by the PINN. The used training data set covers the time between the outbreak of the pandemic in Germany and the last week of the year 2021.We obtain a two-step or hybrid approach, as the PINN is then used to generate a future COVID-19 outbreak scenario describing a possibly next pandemic wave. The week at which the prediction starts is chosen in mid-April 2022.

On the advantages of nonstandard finite differences discretizations for differential problems

On the advantages of nonstandard finite differences discretizations for differential problems

Solution of convection-diffusion problems using fourth order adaptive cubic spline method

In this paper, using adaptive cubic spline, we have suggested a numerical scheme for solving a convection-diffusion problem having layer structure. The numerical scheme is derived with this spline and non-standard finite differences of the first derivative. The tridiagonal solver is used to solve the system of the numerical method. The analysis of convergence of the method is briefly discussed and the fourth order is shown. The numerical results of the examples were tabulated and compared to the existing computational results in order to support the higher accuracy of the proposed numerical scheme.

Solution of convection- diffusion problems using fourth order adaptive cubic spline method

In this paper, using adaptive cubic spline, we have suggested a numerical scheme for solving a convection-diffusion problem having layer structure. The numerical scheme is derived with this spline and non-standard finite differences of the first derivative. The tridiagonal solver is used to solve the system of the numerical method. The analysis of convergence of the method is briefly discussed and the fourth order is shown. The numerical results of the examples were tabulated and compared to the existing computational results in order to support the higher accuracy of the proposed numerical scheme.

Two finite difference methods for a nonlinear BVP arising in physical oceanography

In this paper we define two finite difference methods for a nonlinear boundary value problem on infinite interval. In particular, we report and compare the numerical results for an ocean circulation model obtained by the free boundary approach and a treatment of the problem on the original semi infinite domain by introducing a quasi-uniform grid. In the first case we apply finite difference formulae on a uniform grid and in the second case we use non-standard finite differences on a quasi-uniform grid. We point out how both approaches represent reliable ways to solve boundary value problems defined on semi-infinite intervals. In fact, both approaches overcome the need to define {\it a priori}, or find by trials, a suitable truncated boundary used by the classical numerical treatment of boundary value problems defined on a semi-infinite interval. Finally, the reported numerical results allow to point out how the finite difference method with a quasi-uniform grid is the least demanding approach between the two and that the free boundary approach provides a more reliable formulation than the classical truncated boundary one.

Nonstandard finite differences for a truncated Bratu–Picard model

In this paper, we consider theoretical and numerical properties of a nonlinear boundary-value problem which is strongly related to the well-known Gelfand–Bratu model with parameter λ. When approximating the nonlinear term in the model via a Taylor expansion, we are able to find new types of solutions and multiplicities, depending on the final index N in the expansion. The number of solutions may vary from 0, 1, 2 to ∞. In the latter case of infinitely many solutions, we find both periodic and semi-periodic solutions. Numerical experiments using a non-standard finite-difference (NSFD) approximation illustrate all these aspects. We also show the difference in accuracy for different denominator functions in NSFD when applied to this model. A full classification is given of all possible cases depending on the parameters N and λ.

Nonstandard explicit third-order Runge-Kutta method with positivity property

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) positivity is not ensured when applied to the inhomogeneous linear systems and the same result is regained on nonlinear positivity for this method. Here we mean by positivity that the nonnegativity of the components of the initial vector is preserved. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition to NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, we investigate the positivity property for nonstandard RK3 method when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.

Stability and accuracy of free surface time integration in viscous flows

Geodynamic simulations increasingly rely on models with a true free surface to investigate questions of dynamic topography, tectonic deformation, gravity perturbations, and global mantle convection. However, implementations of free surface boundary conditions have proven challenging from a standpoint of accuracy, robustness, and stability. In particular, time integration of a free surface tends to suffer from a numerical instability that manifests as sloshing surface motions, also known as the “drunken sailor” instability. This instability severely limits stable timestep sizes to those much smaller than can be used in geodynamic simulations without a free surface. Several schemes have been proposed in the literature to deal with these instabilities.Here we analyze the problem of creeping viscous flow with a free surface and discuss the origin of these instabilities. We demonstrate their cause and how existing stabilization schemes work to damp them out. We also propose a new scheme for removing instabilities from free surface calculations. It does not require modifications to the system matrix, nor additional variables, but is instead an explicit scheme based on nonstandard finite differences. It relies on a single stabilization parameter which may be identified with the smallest relaxation timescale of the free surface.Finally, we present numerical results to show the effectiveness of the new approach and discuss the free surface implementation in the open source, community based mantle convection software ASPECT.

Analysis of Some Features of the Immunological System

The communication is devoted to investigation of some properties of a mathematical model describing the response of the immunological system of a vertebrate to infection caused by foreign antigen. The model is formulated as a system of ordinary differential equations [1], [2], [3]. A new numerical algorithm based on non-standard finite differences [4], [5] is proposed. The qualitative properties of the numerical scheme are analyzed. Results of numerical experiments are presented and discussed from computational and biological view points. [1] M. A. Nowak, C. R. Bangham, Population dynamics of immune responses to persistent viruses , Science 272 (5258) (1996) 74--79. [2] R. J. De Boer, A. S. Perelson, Target cell limited and immune control models of HIV infection: a comparison , Journal of Theoretical Biology 190 (3) (1998) 201--214. [3] D. Wodarz, Killer Cell Dynamics: Mathematical and Computational Approaches to Immunology, Springer, New York, 2007 [4] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing, Singapore, 1994 [5] D. T. Dimitrov, H. V. Kojouharov, B. M. Chen-Charpentier, Reliable finite difference schemes with applications in mathematical ecology , in: R. E. Mickens (Ed.), Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing, Singapore, 2005, 249--285

Non standard finite difference scheme preserving dynamical properties

We study the construction of a non-standard finite differences numerical scheme for a general class of two dimensional differential equations including several models in population dynamics using the idea of non-local approximation introduced by R. Mickens. We prove the convergence of the scheme, the unconditional, with respect to the discretization parameter, preservation of the fixed points of the continuous system and the preservation of their stability nature. Several numerical examples are given and comparison with usual numerical scheme (Euler, Runge–Kutta of order 2 or 4) is detailed.

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.

A split-step numerical method for the time-dependent Dirac equation in 3-D axisymmetric geometry

A numerical method is developed to solve the time-dependent Dirac equation in cylindrical coordinates for 3-D axisymmetric systems. The time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration, while the wave function is discretized spatially on a uniform grid. The longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses Poisson's integral solution, which allows to implement the radial symmetry of the wave function. The latter is evaluated on a time staggered mesh by using Hermite polynomial interpolation and by performing the integration analytically. The cylindrical coordinate singularity problem at r=0 is circumvented by this method as the integral is well-defined at the origin. The resulting scheme is reminiscent of non-standard finite differences. In the last step of the splitting, the remaining equation has a solution in terms of a time-ordered exponential, which is approximated to a higher order than the time evolution scheme. We study the time evolution of Gaussian wave packets, and we evaluate the eigenstates of hydrogen-like systems by using a spectral method. We compare the numerical results to analytical solutions to validate the method. In addition, we present three-dimensional simulations of relativistic laser–matter interactions, using the Dirac equation.

Nonstandard finite difference schemes based on Green's function formulations for reaction–diffusion–convection systems

Nonstandard finite differences (NSFD) schemes can improve the accuracy and reduce computational costs of traditional finite-difference schemes. However, the development of these schemes is based on analytical solutions and ad hoc rules. In this paper, we derive a NSFD scheme based on Green's function formulations for reaction–diffusion–convection systems. The formulation of the NSFD scheme is divided into three stages. In the first one, the domain of the original boundary value problem is decomposed into N subdomains. In the second stage, for each subdomain, integral formulations based on Green's functions are derived for each subdomain. Finally, the resulting integrals are approximated by means of quadrature rules. The proposed NSFD scheme based on Green's function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation (i.e., avoiding the use of heuristic rules), it also exhibits global approximation orders of O(h2) and leads to smaller approximation errors with respect to standard FD schemes. Numerical simulations on a catalytic particle model and a benchmark tubular reactor model are used for illustrating the accuracy and performance of the proposed NSFD scheme as compared to standard finite differences schemes.

Non-standard finite-differences schemes for reaction–diffusion equations in curvilinear coordinates

In this work, we extend our previous results (Alvarez-Ramirez, J., Valdes-Parada, F., Alvarez, J., & Ochoa-Tapia, A. (2007). A Green’s function formulation for finite-differences schemes. Chemical Engineering Science, 62, 3083–3091) to the case of reaction–diffusion equations in cylindrical and spherical coordinates. Contrary to standard finite-differences (FD) schemes, the Green’s function formulation incorporates domain curvature effects by weighting the central differences with non-linear metrics, indeed, logarithmic distance for cylindrical and inverse distance for spherical coordinates. It is shown that the Green’s function formulation can be reduced to standard FD schemes for small values of the grid spacing. Numerical simulations show that the modifications introduced improve the performance of traditional FD schemes, specially for coarse grids.

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Reaction–diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. In this work we design, in a systematic way, non-standard finite-differences (FD) schemes for a class of reaction–diffusion equations of the form 1 x σ d d x ( x σ d u d x ) = f ( u ( x ) ) , x ∈ [ 0 , 1 ] , where σ is the shape power that accounts for the complexity of the domain geometry. The proposed FD scheme, that is derived from a Green’s function formulation, replicates the underlying geometry and reduces to traditional FD schemes for sufficiently small values of the grid spacing. Numerical results show that the non-standard FD scheme offers smaller approximation errors with respect to traditional schemes, specially for coarse grids.

High accuracy simulations of optical propagation in nanophotonic devices

We introduce a new high accuracy second-order finite-difference time-domain (FDTD) algorithm based on non-standard finite differences, and use it to investigate light propagation in subwavelength structures such as photonic crystals.