Extended Kung–Traub Methods for Solving Equations with Applications

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Novel Symmetric Numerical Methods for Solving Symmetric Mathematical Problems

The mathematical model for many problems is arising in different industries of natural science, basically formulated using differential, integral and integro-differential equations. The investigation of these equations is conducted with the help of numerical integration theory. It is commonly known that a class of problems can be solved by applying numerical integration. The construction of the quadrature formula has a direct relation with the computation of definite integrals. The theory of definite integrals is used in geometry, physics, mechanics and in other related subjects of science. In this work, the existence and uniqueness of the solution of above-mentioned equations are investigated. By this way, the domain has been defined in which the solution of these problems is equivalent. All proposed four problems can be solved using one and the same methods. We define some domains in which the solution of one of these problems is also the solution of the other problems. Some stable methods with the degree p<=8 are constructed to solve some problems, and obtained results are compared with other known methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. On the intersection of multistep and hybrid methods have been constructed multistep methods and have been proved that these methods are more exact than others. And also has been shown that, hybrid methods constructed here are more exact than Gauss methods. Noted that constructed here hybrid methods preserves the properties of the Gauss method.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

Exponential mean-square stability properties of stochastic linear multistep methods

The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

Multistep Methods of the Hybrid Type and Their Application to Solve the Second Kind Volterra Integral Equation

There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the construction methods, with the best properties of the intersection of them. We also show some connection of the methods constructed here with the methods which are using solving of the initial-value problem for ODEs of the first order. Some of the constructed methods have been applied to solve model problems. A formula is proposed to determine the maximal values of the order of accuracy for the stable and unstable methods, constructed here. Note that to construct the new methods, here we propose to use the system of algebraic equations which allows us to construct methods with the best properties by using the minimal volume of the computational works at each step. For the construction of more exact methods, here we have proposed to use the multistep second derivative method, which has comparisons with the known methods. We have constructed some formulas to determine the maximal order of accuracy, and also determined the necessary and sufficient conditions for the convergence of the methods constructed here. One can proved by multistep methods, which are usually applied to solve the initial-value problem for ODE, demonstrating the applications of these methods to solve Volterra integro-differential equations. For the illustration of the results, we have constructed some concrete methods, and one of them has been applied to solve a model equation.

Comparative Review on Computational Performance of Multistep Schemes in Solving One-Dimensional Linear Wave Equation

Among several numerical methods used to solve the hyperbolic model of the linear wave equation, single-step algorithms can be the more popular ones. However, these algorithms are time-consuming while incurring numerical inaccuracy. Thus, multistep methods can be a suitable option as it has a high order of accuracy. This study aims to investigate and compare the computational performance of these multistep schemes in solving hyperbolic model based on one-dimensional linear wave equation. The techniques studied in this paper comprise the two-step Lax-Wendroff method, MacCormack method, second-order upwind method, Rusanov-Burstein-Mirin method, Warming-Kutler-Lomax method, and fourth-order Runge-Kutta method. Finite difference method is applied in discretisation. Our simulation found that although higher-order multistep methods are more stable than single-step algorithm, they suffer numerical diffusion. The two-step Lax-Wendroff method outperforms other schemes, although it is relatively simple compared with the other three and four steps schemes. The second-order upwind method is attractive as well because it is executable even with a high Courant number.

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.