Search

The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice. [ABSTRACT FROM AUTHOR]

Nonlinear coupled reaction-diffusion systems often arise in cooperative processes of chemical kinetics and biochemical reactions. Owing to these potential applications, this article presents an efficient and simple meshless approximation scheme to analyze the solution behavior of a two-dimensional coupled Brusselator system. On considering radial basis functions in the localized settings, meshless shape functions owing Kronecker delta function property are constructed to discretize the spatial derivatives in the time-dependent partial differential equation (PDE). A system of first-order ordinary differential equations (ODEs), obtained after spatial discretization, is then integrated in time via a high-order ODE solver. The proposed scheme's convergence, stability, and efficiency are theoretically established and numerically verified on several benchmark problems. The outcomes verify reliability, accuracy, and simplicity of the proposed scheme against the available methods in the literature. Some recommendations are made regarding time-step size under different node distributions and RBFs. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical behavior of the radial basis functions least-squares collocation (RBF-LSC) method of lines (MoL) for solving surface diffusion problems, building upon the theoretical analysis presented in [Chen et al. SIAM J. Numer. Anal. 61(3), 1386–1404 (2023)]. Specifically, we examine the impact of the oversampling ratio, defined as the number of collocation points used over the number of RBF centers for quasi-uniform sets, on the stability of the eigenvalues, time-stepping sizes taken by Runge–Kutta methods, and overall accuracy of the method. By providing numerical evidence and insights, we demonstrate the importance of the oversampling ratio for achieving accurate and efficient solutions with the RBF-LSC-MoL method. Our results reveal that the oversampling ratio plays a critical role in determining the stability of the eigenvalues, and we provide guidelines for selecting an optimal oversampling ratio that balances accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion's invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical approach, which is based on the Method of Lines. This method semi-discretizes the problem and produces a system of ordinary differential equations (ODEs) in time. To solve this system, a stiff solver, BDF2, is used, which yields very precise results. The linearization is handled by the Taylor series method. To validate the numerical method, various test examples are considered. These formulas find extensive applications across various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

Numerical methods for computing the solutions of Markov backward stochastic differential equations (BSDEs) driven by continuous-time Markov chains (CTMCs) are explored. The main contributions of this paper are as follows: (1) we observe that Euler-Maruyama temporal discretization methods for solving Markov BSDEs driven by CTMCs are equivalent to exponential integrators for solving the associated systems of ordinary differential equations (ODEs); (2) we introduce multi-stage Euler–Maruyama methods for effectively solving "stiff" Markov BSDEs driven by CTMCs; these BSDEs typically arise from the spatial discretization of Markov BSDEs driven by Brownian motion; (3) we propose a multilevel spatial discretization method on sparse grids that efficiently approximates high-dimensional Markov BSDEs driven by Brownian motion with a combination of multiple Markov BSDEs driven by CTMCs on grids with different resolutions. We also illustrate the effectiveness of the presented methods with a number of numerical experiments in which we treat nonlinear BSDEs arising from option pricing problems in finance. [ABSTRACT FROM AUTHOR]

Viscous Burger’s equation is one of the most celebrated models with immense applications cutting across all aspects of mathematical physics. Thus, the present communication makes use of the non-central formula infused in the method of lines coupled with Runge-Kutta spatial discretization to computationally and efficiently treat the coupled system of viscous Burger’s equations. Further, we numerically tested the derived schemes of the governing model amidst suitable initial and boundary data. In fact, we eventually analyzed the effectiveness of the method via L2 and L∞ norms on some test problems and found it to be robust; indeed, a comparison of the current scheme with some notable approaches in the literature has been established, which are realized to be in perfect agreement. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O (Δ t + h ¯ 4) , where Δ t is the time step and h ¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. [ABSTRACT FROM AUTHOR]

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N-storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods. [ABSTRACT FROM AUTHOR]

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and i i) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized- α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time "slices" in an attempt to reduce the computational cost of the space-time discretizations. We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6] , the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies. [ABSTRACT FROM AUTHOR]

Numerical schemes for advection-dominated transport problems are are evaluated in a comparative study. Explicit and implicit finite difference methods are analyzed together with a global random walk algorithm in the frame of a splitting procedure. The efficiency of the methods with respect to the control of the numerical diffusion is investigated numerically on one-dimensional problems with constant coefficients and two-dimensional problems with variable coefficients consisting of realizations of space-random functions.

A recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid, steady conservation laws [43,45] is extended to the unsteady case. Central to the framework is an optimization problem which simultaneously computes a discontinuity-aligned mesh and the corresponding high-order approximation to the flow, which provides nonlinear stabilization and a high-order approximation to the solution. This work extends the implicit shock tracking framework to the case of unsteady conservation laws using a method of lines discretization via a diagonally implicit Runge-Kutta method by “solving a steady problem at each timestep”. We formulate and solve an optimization problem that produces a feature-aligned mesh and solution at each Runge-Kutta stage of each timestep, and advance this solution in time by standard Runge-Kutta update formulas. A Rankine-Hugoniot based prediction of the shock location together with a high-order, untangling mesh smoothing procedure provides a high-quality initial guess for the optimization problem at each time, which results in rapid convergence of the sequential quadratic programing (SQP) optimization solver. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover the design accuracy of the Runge-Kutta scheme. We demonstrate this framework on a series of inviscid, unsteady conservation laws in both one- and two- dimensions. We also verify that our method is able to recover the design order of accuracy of our time integrator in the presence of a strong discontinuity.

In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option. [ABSTRACT FROM AUTHOR]

In this study, the advection–dispersion equation with decay is numerically solved by the finite difference-based method of lines (FD-MOL) to simulate groundwater radionuclide transport. Finite difference orders of 1,2,...,8 are used for spatial approximation, while the linearly implicit Euler scheme is employed adaptively for temporal discretization. Four different problems are investigated, and results show that FD-MOL provides accurate and stable numerical solutions. Coarse temporal grids can be utilized implicitly, for instance, a maximum step of 1000 years with 400 spatial nodes yields RMS errors of 7.508 × 10−6, 7.395 × 10−5 and 7.705 × 10−6 in 92 234 U , 90 230 Th and 88 226 Ra normalized concentrations, respectively, for the decay chain problem. [ABSTRACT FROM AUTHOR]

Nowadays, network structures are found in many natural and engineered systems, e.g., river networks, microchannel networks, plant roots, human blood vessels, etc. Therefore, providing efficient methods for modeling phenomena such as diffusion, advection, etc. is very practical. One of the most common tools for modeling this phenomena is numerical modeling, as mathematics software is well-developed and powerful nowadays. In this research, a new approach called Equation-Oriented Modeling has been presented. In this approach each branch of the network has its own differential equation, and these branches are connected or coupled by boundary conditions. In other words, unlike classical modeling, EOM does not solve through the discretization of the partial differential equation in the whole domain of the network, while in this approach, each branch of the network has its own differential equation with its own specific diffusion coefficient and cross section area, then the problem is solved as a system of PDE. The main point of EOM is to formulate a physical problem in the network into a system of differential equations, which is finally solved by the Method of Lines. MOL is an efficient computational method used to solve partial differential equations or PDE systems. MOL is generally implemented in two steps, in the first step spatial derivatives are replaced by algebraic approximation. In the second step, the ordinary differential equation system is integrated with respect to time using any method, for example, in this research, we use the Runge-Kutta 4th order method. EOM was implemented to solve the diffusion equation in three types of networks, including tree-shaped and loop network. Then modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The model had reasonable results in the boundaries and branches according to the boundary conditions, loading and concentration functions, as well as the continuity of concentrations and loading by diffusion in the output results was reasonable. The boundary conditions that apply at the intersections of the branches include the continuity of concentration and the continuity of loading due to the diffusion phenomenon. The results of test case 3 were compared with another numerical model for validation, and three types of Error Parameters were calculated at different times between these two models. R-Squared (R²) was calculated in path (1-2-3-5-9), and its value was 0.99-1, which was the optimal value. This coefficient shows that the results of the EOM and the other numerical model has the same trend. Then, RMSE and MAE were also calculated and their values were approximately zero for all times. The modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The first advantage of the EOM approach is that the choice of terms in the differential equation is left to the user rather than the software developer, so that a wider range of phenomena can be modeled and the effects of different terms can be seen in the modeling. The second advantage of this approach over classical modeling is that the equations are available to the user as tools and model elements, and modeling complex networks such as tree-shaped, and Loop networks is not as complicated as classical models. The third advantage of EOM is the tools available in mathematical programs for optimization or linking with other programs. Since the heat equation is similar to the diffusion equation, the results of this research can be used for other important topics, such as solving the heat equation in microchannel networks for cooling systems, modeling pollutant transport in river networks, or diffusion modeling of solutes in plant roots. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method.

In this study, we have simulated transient and steady state free convection flow and heat transfer inside a square enclosure filled with a copper-water nanofluid of spherical shape nanoparticles following Tiwari-Das model. The cavity containing an insulated rectangular obstacle of height ranging from 0% to 50% of the cavity side-length. The vertical sides of the enclosure are kept at different temperatures, while the flat sides are assumed to be adiabatic as the obstacle. The combined method of lines/penalty-artificial compressibility technique (MOL-PACT) has been applied to solve the dimensional time dependent mathematical model after converting it into a non-dimensional structure. The combined method of lines/penalty-artificial compressibility technique is recently successfully applied to simulate free convection of MHD fluid in square enclosure with a localized heating. The extension of this promising technique for studying heat transfer of nanofluids is one of the objectives of this paper. Another objective of the study is to inspect the impact of several model parameters such as, the obstacle height, nanoparticles volume-fraction, nanoparticles radius and Rayleigh number on streamlines, temperature distribution and Nusselt number as an expression of heat transfer inside the enclosure. The results have been discussed and shown graphically. Comparisons with former results for related cases in the literature are made and reasonably good agreements are observed.

We analyze the dependence of radiation loss, effective indices and difference of the effective indices of the two modes with similar field distribution propagating in dual-core microstructured fibres as well as their polarization behavior on fibre parameters (air hole diameter, hole separation, distance between guiding cores) and fibre bending. Optimization of the parameters of such fibres using as vector bend sensors is considered.

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent. [ABSTRACT FROM AUTHOR]

There are plenty of applications and analyses for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used asymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which are methods of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces. [ABSTRACT FROM AUTHOR]

In this paper, a class of Volterra fractional partial integro-differential equations (VFPIDEs) with initial conditions is investigated. Here, the well-known method of lines (MOLs) is developed to solve the VFPIDEs. To this end, the VFPIDE is converted into a system of first-order ordinary differential equations (ODEs) in time variable with initial conditions. Then the resulting ODE system is solved by an LN-stable method, such as Radau IIA or Lobatto IIIC. It is proved that the proposed method is LNstable. Also, the convergence of the proposed method is proved. Finally, some numerical examples are given to illustrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Fractional differential equations (FDEs) are used for modeling the natural phenomena and interpretation of many life problems in the fields of applied science and engineering. The mathematical models which include different types of differential equations are used in some fields of applied sciences like biology, diffusion, electronic circuits, damping laws, fluid mechanics, and many others. The derivation of modern analytical or numerical methods for solving FDEs is a significant problem. However, in this article, we introduce a novel approach to generalize Runge Kutta Mechee (RKM) method for solving a class of fifth-order fractional partial differential equations (FPDEs) by combining numerical RKM techniques with the method of lines. We have applied the developed approach to solve some problems involving fifth-order FPDEs, and then, the numerical and analytical solutions for these problems have been compared. The comparisons in the implementations have proved the efficiency and accuracy of the developed RKM method.

In this paper, we develop an iterative method of lines scheme for the numerical solution to the time fractional Richards equation with implicit Neumann boundary conditions, which is an effective tool for describing a process of flow through unsaturated media. A numerical example is provided to show the effectiveness of the presented method for different model parameters and inputs. The method illustrated here can be applied to other types Richards equation with various input functions and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

The article presents the results of studies on the natural vibration frequencies of circular truncated conical shells completely filled with an ideal compressible fluid. The behavior of the elastic structure is described in the framework of classical shell theory, the equations of which are written in the form of a system of ordinary differential equations with respect to new unknowns. Small fluid vibrations are described by the linearized Euler equations, which in the acoustic approximation are reduced to the wave equation with respect to hydrodynamic pressure and written in spherical coordinates. Its transformation to the system of ordinary differential equations is performed in three ways: by the straight line method, by spline interpolation and by the method of differential quadrature. The formulated boundary value problem is solved using the Godunov orthogonal sweep method. The calculation of natural frequencies of vibrations is based on the application of a stepwise procedure and subsequent refinement by the half-division method. The validity of the results obtained is confirmed by their comparison with known numerical-analytical solutions. The efficiency of frequency calculations in the case of using different methods of wave equation transformation is evaluated for shells with different combinations of boundary conditions and cone angles. It is demonstrated that the use of the generalized differential quadrature method provides the most cost-effective solution to the problem with acceptable calculation accuracy. [ABSTRACT FROM AUTHOR]

Development of quick and versatile simulation packages to investigate time dependent heat transfer phenomena in nuclear reactors or steam generators is an ongoing research interest. The work presented herein is an alternative attempt to address this issue employing a single heated channel framework. A homogenous equivalent mixture is assumed for the two phase flow inside the channel. The sectionalized compressible formalism is further refined and the resultant PDEs are cast into a nodalized layout employing the method of lines as a reduction scheme. Simulation results are presented for the transient heat transfer inside a coolant channel of a nuclear reactor core wherein the incumbent heat flux is affected through an inherent thermo-neutronic feedback mechanism. The overall model reduction strategy likewise provides a suitable platform for the purpose of stability analysis or control synthesis practices. [ABSTRACT FROM AUTHOR]

Physical constraints must be taken into account in solving partial differential equations (PDEs) in modeling physical phenomenon time evolution of chemical or biological species. In other words, numerical schemes ought to be devised in a way that numerical results may have the same qualitative properties as those of the theoretical results. Methods with monotonicity preserving property possess a qualitative feature that renders them practically proper for solving hyperbolic systems. The need for monotonicity signifies the essential boundedness properties necessary for the numerical methods. That said, for many linear multistep methods (LMMs), the monotonicity demands are violated. Therefore, it cannot be concluded that the total variations of those methods are bounded. This paper investigates monotonicity, especially emphasizing the stepsize restrictions for boundedness of A-BDF methods as a subclass of LMMs. A-stable methods can often be effectively used for stiff ODEs, but may prove inefficient in hyperbolic equations with stiff source terms. Numerical experiments show that if we apply the A-BDF method to Sod's problem, the numerical solution for the density is sharp without spurious oscillations. Also, application of the A-BDF method to the discontinuous diffusion problem is free of temporal oscillations and negative values near the discontinuous points while the SSP RK2 method does not have such properties.

In this study, we have simulated transient and steady state free convection flow and heat transfer inside a square enclosure filled with a copper-water nanofluid of spherical shape nanoparticles following Tiwari-Das model. The cavity containing an insulated rectangular obstacle of height ranging from 0% to 50% of the cavity side-length. The vertical sides of the enclosure are kept at different temperatures, while the flat sides are assumed to be adiabatic as the obstacle. The combined method of lines/penalty-artificial compressibility technique (MOL-PACT) has been applied to solve the dimensional time dependent mathematical model after converting it into a non-dimensional structure. The combined method of lines/penalty-artificial compressibility technique is recently successfully applied to simulate free convection of MHD fluid in square enclosure with a localized heating. The extension of this promising technique for studying heat transfer of nanofluids is one of the objectives of this paper. Another objective of the study is to inspect the impact of several model parameters such as, the obstacle height, nanoparticles volume-fraction, nanoparticles radius and Rayleigh number on streamlines, temperature distribution and Nusselt number as an expression of heat transfer inside the enclosure. The results have been discussed and shown graphically. Comparisons with former results for related cases in the literature are made and reasonably good agreements are observed. [ABSTRACT FROM AUTHOR]

In this paper, a numerical technique is proposed to solve a two-dimensional coupled Burgers' equation. The two-dimensional Cole–Hopf transformation is applied to convert the nonlinear coupled Burgers' equation into a two-dimensional linear diffusion equation with Neumann boundary conditions. The diffusion equation with Neumann boundary conditions is semi-discretized using MOL in both x and y directions. This process yielded the system of ordinary differential equations in the time variable. Multistep methods namely backward differentiation formulas of order one, two and three are employed to solve the ode system. Efficiency and accuracy of the proposed methods are verified through numerical experiments. The proposed schemes are simple, accurate, efficient and easy to implement. [ABSTRACT FROM AUTHOR]

The Method of Lines in combination with impedance/admittance and field transformation is used to analyze electromagnetic waves. The used cases are waveguiding structures in microwave technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the FD are permanently linked to the steps of the fully vectorial impedance/admittance and field transformation, so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built into the impedance/admittance and field transformation in the past. However, the degree of improvement due to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an extension of the family of built-in methods is proposed—with the possibility of being able to build any known numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices for the impedance/admittance and field transformation serve as the building site. To illustrate the procedure, some different methods are integrated into the GTL solution. The efficiency of the solutions is tested on some test structures and compared with each other and with existing solutions. The relevant waveguide specifics were discussed. Initial systematics and recommendations for users were derived. [ABSTRACT FROM AUTHOR]

An algorithm of axisymmetric unbranched soft shells nonlinear dynamic behaviour problems solution is suggested in the work. The algorithm does not impose any restrictions on deformations or displacements range, material properties, conditions of fixing or meridian form of the structure. Mathematical statement of the problem is given in vector-matrix form and includes system of partial differential equations, system of additional algebraic equations, structure segments coupling conditions, initial and boundary conditions. Partial differential equations of motion are reduced to nonlinear ordinary differential equations using method of lines. Obtained equation system is differentiated by calendar parameter. As a result problem solution is reduced to solving two interconnected problems: quasilinear multipoint boundary problem and nonlinear Cauchy problem with right-hand side of a special form. Features of represented algorithm using in application to the problems of soft shells dynamics are revealed at its program realization and are described in the work. Three- and four-point finite difference schemes are used for acceleration approximation. Algorithm testing is carried out for the example of hinged hemisphere of neo-hookean material dynamic inflation. Influence of time step and acceleration approximation scheme choice on solution results is investigated.

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

This article presents a new numerical model with nodes within the interval of eight-step for numerical simulation of a class of variable coefficient elliptic partial differential equations in the two-dimensional domain. The method is developed via the principle of collocation and interpolation techniques using Hermite polynomial as the basis function. The main classical model and its derivatives generated from the continuous function are then united together to form the required Classical Eight-step Model (CEM). The analysis of the CEM was investigated and shows that it satisfies the conditions for convergence with an algebraic order of nine. The CEM is applied to solve the semi-discretized elliptic partial differential equations with a variable coefficient arising from the discretization of one of their spatial variables. The performance of the CEM was established with six test problems. The approximate solution generated using CEM is compared with the analytical solution of the problems and other existing methods in the literature. Numerical illustrations demonstrate that the method is convergent and highly accurate. The advantages of CEM over other existing methods reveal the accuracy and efficiency of the CEM as depicted in curves.

Abstract In this work, we develop an efficient numerical scheme based on the method of lines (MOL) to investigate the interesting phenomenon of collisions and reflections of optical solitons. The established scheme is of second order in space and of fourth order in time with an explicit nature. We deduce stability restrictions using the von Neumann stability analysis. We consider a ( 2 + 1 ) $(2+ 1)$ -dimensional system of a coupled nonlinear Schrödinger equation as a general model of nonlinear Schrödinger-type equations. We consider several numerical experiments to demonstrate the robustness of the scheme in capturing many scenarios of collisions and reflections of the optical solitons related to nonlinear Schrödinger-type equations. We verify the scheme accuracy through computing the conserved invariants and comparing the present results with some existing ones in the literature.

This paper deals with the numerical solutions of a general class of one-dimensional nonlinear partial differential equations (PDEs) arising in different fields of science. The nonlinear equations contain, as special cases, several PDEs such as Burgers equation, nonlinear-Schrödinger equation (NLSE), Korteweg–De Vries (KDV) equation, and KdV–Schrödinger equations. Inspired by the method of lines, an RBF-FD approximation of the spatial derivatives in terms of local unknown function values, converts the nonlinear governing equations to a system of nonlinear ordinary differential equations(ODEs). Then, a fourth-order Runge–Kutta method is proposed to solve the resulting nonlinear system of first-order ODEs. For the RBF-FD approximation of derivatives, three kinds of different basis are investigated, and it is shown that the polynomial basis gives the highest accuracy. Solitary wave solutions form a special class of solutions of the model equations considered in this paper. The results reveal that the proposed method simulates this kind of solutions with high accuracy. Also, the method is able to simulate the collision of two soliton solutions, provided they are initially located far enough from each other. [ABSTRACT FROM AUTHOR]

In this article, an efficient method for approximating the solution of the generalized Burgers-Huxley (gB-H) equation using a multiquadric quasi-interpolation approach is considered. This method consists of two phases. First, the spatial derivatives are evaluated by MQ quasi-interpolation, So the gB-H equation is reduced to a nonlinear system of ordinary differential equations. In phase two, the obtained system is solved by using ODE solvers. Numerical examples demonstrate the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems. [ABSTRACT FROM AUTHOR]

In this article, it is aimed to address one of Ostwald-de Waele fluid problems that either, has not been addressed or very little focused on. Considering the impacts of heat involving in the Non-Newtonian flow, a variant thickness of the disk is additionally considered which is governed by the relation z = a (r/RO+1)-m. The rotating Non-Newtonian flow dynamics are represented by the system of highly nonlinear coupled partial differential equations. To seek a formidable solution of this nonlinear phenomenon, the application of the method of lines using von Kármán’s transformation is implemented to reduce the given PDEs into a system of nonlinear coupled ordinary differential equations. A numerical solution is considered as the ultimate option, for such nonlinear flow problems, both closed-form solution and an analytical solution are hard to come by. The method of lines scheme is preferred to obtain the desired solution which is found to be more reliable and in accordance with the required physical expectation. Eventually, some new marvels are found. Results indicate that, unlike the flat rotating disk, the local radial skin friction coefficients and tangential decrease with the fluid physical power-law exponent increases, the peak in the radial velocity rises which is significantly distinct from the results of a power-law fluid over a flat rotating disk. The local radial skin friction coefficient increases as the disk thickness index increases, while local tangential skin friction coefficient decreases, the local Nusselt number decrease, both the thickness of the velocity and temperature boundary layer increase.