Computational and mathematical models meet heterogeneous computing

The Journal of Supercomputing -     

High order Bessel fitting methods for the numerical integration of the Schrödinger equation

In this paper we show how to construct explicit multistep algorithms for an accurate and efficient numerical integration of the radial Schr?dinger equation. The proposed methods are Bessel fitting, that is to say, they integrate exactly any linear combination of Bessel and Newman functions and ordinary polynomials. They are the first of the like methods that can achieve any order.     

A trigonometrically-fitted method with two frequencies,one for the solution and another one for the derivative

In trigonometrically-fitted methods the determination of the parameter (usually known as the frequency) is a critical issue, as was shown in the article by H. Ramos and J. Vigo-Aguiar [Applied Mathematics Letters, 23 (2010) 1378–1381]. If the frequency estimation relies on the vanishing of the principal term of the local truncation error, then the first derivative is present in the formula for approximating the parameter. This requires the use of a procedure for approximating the first derivative. For this purpose we use another trigonometrically-fitted formula with a second parameter (different from the frequency of the principal method and also different from the frequency of the true solution). We describe how to approximate both parameters on each step and present different experiments concerning these questions. The numerical results indicate the good performance of the strategy.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

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.     

Adapted BDF Algorithms: Higher-order Methods and Their Stability

We present BDF type formulas of high-order (4, 5 and 6), 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. For A = 0, the new formulas reduce to the classical BDF formulas. Theorems of the local truncation error reveal the good behavior of the new methods with stiff problems. Plots of their 0-stability regions in terms of the eigenvalues of the parameter A h are provided. Plots of their absolute stability regions that include the whole of the negative real axis are provided. The weights of the method usually require the evaluation of a matrix exponential. However, if the dimension of the matrix is large, we shall not perform this calculus and shall only approximate those coefficients once. Numerical examples underscore the efficiency of the proposed codes, especially when one is integrating stiff oscillatory problems.