J. C. Jimenez

H. de la Cruz, R. J. Biscay, J. C. Jimenez et al.

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.

C. M. Mora, H. A. Mardones, J. C. Jimenez et al.

We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by integrating the system of coupled SDEs that describes the evolution of the norm of $X_t$ and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from $0$. The scheme also has linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients,and one-half strong order of convergence.

F. Carbonell, Y. Iturria-Medina, J. C. Jimenez

The combination of the multiple shooting strategy with the generalized Gauss-Newton algorithm turns out in a recognized method for estimating parameters in ordinary differential equations (ODEs) from noisy discrete observations. A key issue for an efficient implementation of this method is the accurate integration of the ODE and the evaluation of the derivatives involved in the optimization algorithm. In this paper, we study the feasibility of the Local Linearization (LL) approach for the simultaneous numerical integration of the ODE and the evaluation of such derivatives. This integration approach results in a stable method for the accurate approximation of the derivatives with no more computational cost than the that involved in the integration of the ODE. The numerical simulations show that the proposed Multiple Shooting-Local Linearization method recovers the true parameters value under different scenarios of noisy data.

J. C. Jimenez, C. Mora, M. Selva

In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme.