J. Martín-Vaquero

Exponential fitting BDF-Runge-Kutta algorithms

In other papers, the authors presented exponential fitting methods of BDF type. Now, these methods are used to derive some BDF–Runge–Kutta type formulas (of second-, third- and fourth-order), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. Different procedures to find the parameter of the method are proposed, using these techniques there will not be necessary to compute the exponential matrix at each step, even when nonlinear problems are integrated. Numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

Second-order stabilized explicit Runge–Kutta methods for stiff problems

Abstract Stabilized Runge–Kutta methods (they have also been called Chebyshev–Runge–Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge–Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10 5 ). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.

Numerical algorithms for diffusion–reaction problems with non-classical conditions

Abstract Parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years. In this paper new high-order algorithms for the linear diffusion–reaction problem are derived. The convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion–reaction equations with these non classical conditions.

Exponential fitting BDF algorithms and their properties

We present two families of explicit and implicit BDF formulas, capable of the exact integration (with only round-off errors) of differential equations whose solutions belong to the space generated by the linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. Those methods are suitable for stiff and highly oscillatory problems, then we will study their properties of local truncation error and stability. Plots of their 0-stability regions in terms of Ah are provided. Plots of their regions of absolute stability that include all the negative real axis in the implicit case are provided. Exponential fitting algorithms depend on a parameter Ah, how can we find this parameter is a big question, here we show different ways to find a good parameter. Finally, numerical examples show the efficiency of the proposed codes, specially when we are integrating stiff problems where the jacobian of the function has complex eigenvalues or problems where the jacobian has positive eigenvalues but the solutions of the problems have not positive exponentials.

Exponential fitted Gauss, Radau and Lobatto methods of low order

Several exponential fitting Runge-Kutta methods of collocation type are derived as a generalization of the Gauss, Radau and Lobatto traditional methods of two steps. The new methods are capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. A different procedure to find the parameter of the method is proposed. The variable step Radau method of two stages is derived. Finally, numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

CAS and real life problems to learn basic concepts in Linear Algebra course

In this paper we examine the effect of the use of a Computer Algebra System (CAS) to solve interdisciplinary problems. We have been working for some years in the application of mathematics to different engineering problems, the main reason was to improve the motivation of our students and to make possible for them to connect and apply what they learn in Linear Algebra class to other engineering subjects. We describe a case study with students of Electromechanical and Mechanical Engineering from the Coimbra Institute of Engineering and from Geological and Industrial Engineering students from the University of Salamanca in a Linear Algebra course. The study involves the application of digital image processing to develop the basic skills of Linear Algebra. Furthermore, we include the results of a pre‐test and post‐test control group study using that application in the classroom. The results indicate that the experiences were very enriching for the students in different subjects. We tested at different centers from two different countries with similar results.

Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model

Abstract In the present manuscript, higher-order methods are derived to solve a SEIR model for malware propagation. They are obtained using extrapolation techniques combined with nonstandard finite difference (NSFD) schemes used in Jansen and Twizell (2002). Thus, the new algorithms are more efficient computationally, and are dynamically consistent with the continuous model. Later, different procedures are considered to control the error in the discrete schemes. Numerical experiments are provided to illustrate the theory, and for the comparison of the different strategies in the adaptation of the variable step length.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

A Galerkin method for two-dimensional hyperbolic integro-differential equation with purely integral conditions

The present paper is devoted to the solution for two-dimensional hyperbolic integro-differential equations subject to purely integral conditions. First, it is demonstrated the existence and uniqueness of the solution under certain conditions. For the numerical approach, a Galerkin method based on least squares is proposed. The numerical examples illustrate the technique and show the efficiency of the proposed method.

Efficient high-order finite difference methods for nonlinear Klein-Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein-Gordon equation

In this paper, the problem of solving some nonlinear Klein-Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein-Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs.