Finite volume method for a coupled Stokes-Darcy problem

In this paper we propose a finite volume method to solve the coupled Stokes-Darcy problem using steady Stokes equations for the fluid region and Darcy equations for the porous region. At the contact interface between the fluid region and the porous media we imposed two conditions. The first one is the normal continuity of the velocity, while the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use schwarz domain decomposition method. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the prop- erties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method

Interval extension of the three-step Kung and Traub's method

In this paper we produce an interval extension of the three-step Kung and Traub's method for solving nonlinear equations‎. ‎Furthermore‎, ‎the convergence analysis of the new method is discussed and this method is compared to already present methods‎.

New optimized domain decomposition order 4 method(OO4) applied to reaction advection diffusion equation

The purpose of this work is the study of a new approach of domain decomposition method, the optimized order 4 method(OO4), to solve a reaction advection diusion equation. This method is a Schwarz waveform relaxation approach extending the known OO2 idea. The OO4 method is a reformulation of the Schwarz algorithm with specific conditions at the interface. This condition are a dierential equation of order 1 in the normal direction and of order 4 in the tangential direction to the interface resulting of artificial boundary conditions. The obtained scheme is solved by a Krylov type algorithm. The main result in this paper is that the proposed OO4 algorithm is more robust and faster than the classical OO2 method. To confirm the performance of our method , we give several numerical test-cases.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

Accurate Multistep Multi-derivative Collocation Methods Applied to Chaotic Systems

Accurate multistep multi-derivative collocation methods are derived for the numerical integration of chaotic systems. These methods have high order of accuracy, with A-stable regions of absolute stability. Although the calculation of the third derivative terms in the methods is relatively high compared to the rst and second derivative terms, the advantage gained makes them suitable for solving chaotic system of equations with large eigenvalues. The stability characteristic properties and order of accuracy of the methods are studied. Figurative comparisons of the solution curves obtained are in good agreement with the exact solutions which demonstrate practically the eectiveness of the proposed methods.

Operational formula for the generalized Bessel matrix polynomials

In this paper, we propose to give some operational formula of the generalized Bessel matrix polynomials (GBMPs) using the difference operators. Some special cases of the main results are also established.

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

The contractivity-preserving 2- and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17--39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector, Hermite- Birkhoff--Obrechkoff series methods, denoted by HBO(d,k,p), k=4,5,6,7, with nonnegative coefficients for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generally higher upper bound \(p_u\) of order \(p\) of HBO(d,k,p) for a given d. Their stability regions have generally a good shape and grow generally with decreasing \(p-d\). A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has maximum order 14 based on the CP conditions compares satisfactorily with Adams--Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error, relative energy error and 10000 periods of integration. The coefficients of HBO(6,4,14) are listed in the appendix.

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Second derivative Runge-Kutta collocation methods for the numerical solution of sti system of rst order initial value problems in ordinary dierential equations are derived and studied. The inclusion of the second derivative terms enabled us to derive a set of methods which are convergent with large regions of absolute stability. Although the implementation of the methods remains iterative in a precisely dened way, the advantage gained makes them suitable for solving sti system of equations with large Lipschitz constants. The derived methods are illustrated by the applications to some test problems of sti system found in the literature and the numerical results obtained conrm the potential of the second derivative methods.

The numerical integration of stiff systems using stable multistep multiderivative methods

In this paper we describe the construction of stable multistep multiderivative methods designed for continuous numerical integration of stiff systems of initial value problems in ordinary dierential equations. These methods are obtained based on the multistep collocation technique, which are shown to be A-stable, convergent with large regions of absolute stability. They are suitable for solving stiff systems of initial value problems with large eigenvalues lying close to the imaginary axis. Numerical experiments illustrate the behaviour of the methods, which show that they are competitive with stiff integrators that are known to have strong stability characteristic properties. Comparison of the solution curves obtained is in good agreement with the exact solutions which demonstrate the reliability and usefulness of the methods.

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

A new four-step implicit linear sixth algebraic order method with vanished phase-lag and its first derivative is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schr¨odinger equation and related problems. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.