Some stiffly stable second derivative continuous linear multistep methods with a hybrid point for stiff IVPS in ODEs

Based on Gear¡¦s fixed step size backward differentiation methods, Gear (1968), second derivative continuous linear multistep methods with an off-step point are presented. This type of methods provides a means of bypassing the order barriers imposed by Dahlquist (1963) and also provides continuous solutions of IVPs in ODEs. The stiff stability of these methods is determined by using the boundary locus. Instability sets in at k ƒ­10 . Numerical results of the methods solving a non-linear and a linear stiff initial value problems in ordinary differential equations are compared to that of the state -of -the-art code, ODE 15s in MATLAB

On Extrapolated Multirate Methods

In this manuscript we construct extrapolated multirate discretization methods that allow to efficiently solve problems that have components with different dynamics. This approach is suited for the time integration of multiscale ordinary and partial differential equations and provides highly accurate discretizations. We analyze the linear stability properties of the multirate explicit and linearly implicit extrapolated methods. Numerical results with multiscale ODEs illustrate the theoretical findings

DERIVATION OF A CLASS OF HYBRID ADAMS MOULTON METHOD WITH CONTINOUS COEFFICIENTS

This research work is focused on the derivation of both the continuous and discrete models of the hybrid Adams Moulton method for step number k =1 and k = 2. These formulations incorporate both the off – grid interpolation and off- grid collocation schemes. The convergence analysis reveals that derived schemes are zero stable, of good order and error constants which by implication shows that the schemes are consistent. Keywords: Hybrids Schemes, Adams Methods, Linear K–Step Method, Consistency, Zero Stabl

A new one-step method with three intermediate points in a variable step-size mode for stiff differential systems.

[EN]This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, in particular, in problems derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder in an article on problems of chemical kinetics (Hirschfelder, Proc Natl Acad Sci USA 38:235–243, 1952). The techniques of interpolation and collocation are used in the construction of the scheme. We consider a suitable polynomial to approximate the theoretical solution of the problem under consideration. The basic properties of the new scheme are analyzed. An embedded strategy is adopted to formulate the proposed scheme in a variable stepsize mode to get better performance. Finally, some models of initial-value problems, including ordinary and time-dependent partial differential equations, are solved numerically to assess the performance and efficiency of the proposed technique, with applications to real-world problems

On the convergence of a finite difference scheme for a second order differential equation containing nonlinearly a first derivative

This note is concerned with the convergence of a finite difference scheme to the solution of a second order ordinary differential equation with the right-hand-side nonlinearly dependent on the first derivative. By defining stability as the linear growth of the elements of the inverse of a certain matrix and combining this with consistency, convergence is demonstrated. This stability concept is then interpreted in terms of a root condition

Numerical approximation of second-order boundary value problems via hybrid boundary value method

Hybrid Boundary Value Method (HyBVM) is a new scheme, which is based on Linear Multistep Method (LMM). The HyBVM is the hybrid version of the Boundary Value Methods (BVMs) which are methods derived to overcome the limitations of the LMMs. This new scheme shares the same characteristic with the Runge Kutta method as data are utilized at off-step points. In this work, we apply this method to two second order Boundary Value Problems (BVPs) with mixed boundary conditions and the results are efficient when compared to other BVMs in literature

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

[EN] In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Enhanced Numerov Method for the Numerical Solution of Second Order Initial Value Problems

Numerov method is a multistep numerical method that is used in solving second order differential equations. In this work, we apply this method as a Boundary Value Method (BVM) for the numerical approximation of both linear and nonlinear second order initial value problems. This is achieved by constructing the Numerov method via interpolation and collocation process while utilizing data at off-step points and implementing it as a BVM. On comparing the results obtained from the solved problems, it shows that the method is accurate with high level of convergence to their exact forms and performs better than results from literature