Martina Moccaldi

Adapted numerical methods for advection–reaction–diffusion problems generating periodic wavefronts

Abstract The paper presents an adapted numerical integration for advection–reaction–diffusion problems. The numerical scheme, exploiting the a-priori knowledge of the qualitative behaviour of the solution, gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature. The adaptation is here carried out through the so-called trigonometrical fitting technique for the discretization in space, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit–Explicit (IMEX) method is here employed for the integration in time, based on the first order forward–backward Euler method. The coefficients of the method here introduced rely on unknown parameters which have to be properly estimated. In this work, such an estimate is performed by minimizing the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and some numerical experiments.

On the Employ of Time Series in the Numerical Treatment of Differential Equations Modeling Oscillatory Phenomena

The employ of an adapted numerical scheme within the integration of differential equations shows benefits in terms of accuracy and stability. In particular, we focus on differential equations modeling chemical phenomena with an oscillatory dynamics. In this work, the adaptation can be performed thanks to the information arising from existing theoretical studies and especially the observation of time series. Such information is properly merged into the exponential fitting technique, which is specially suitable to follow the a-priori known qualitative behavior of the solution. Some numerical experiments will be provided to exhibit the effectiveness of this approach.

Stochastic Numerical Models of Oscillatory Phenomena

The use of time series for integrating ordinary differential equations to model oscillatory chemical phenomena has shown benefits in terms of accuracy and stability. In this work, we suggest to adapt also the model in order to improve the matching of the numerical solution with the time series of experimental data. The resulting model is a system of stochastic differential equations. The stochastic nature depends on physical considerations and the noise relies on an arbitrary function which is empirically chosen. The integration is carried out through stochastic methods which integrate the deterministic part by using one-step methods and approximate the stochastic term by employing Monte Carlo simulations. Some numerical experiments will be provided to show the effectiveness of this approach.

Adapted numerical modelling of the Belousov–Zhabotinsky reaction

Adapted numerical schemes for the integration of differential equations generating periodic wavefronts have reported benefits in terms of accuracy and stability. This work is focused on differential equations modelling chemical phenomena which are characterized by an oscillatory dynamics. The adaptation is carried out through the exponential fitting technique, which is specially suitable to follow the apriori known qualitative behavior of the solution. In particular, we have merged this strategy with the information coming from existing theoretical studies and especially the observation of time series. Numerical tests will be provided to show the effectiveness of this problem-oriented approach.

Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction–diffusion problems

Abstract In this paper, an adapted numerical scheme for reaction–diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit–explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.

Efficient boundary corrected Strang splitting

Strang splitting is a well established tool for the numerical integration of evolution equations. It allows the application of tailored integrators for different parts of the vector field. However, it is also prone to order reduction in the case of non-trivial boundary conditions. This order reduction can be remedied by correcting the boundary values of the intermediate splitting step. In this paper, three different approaches for constructing such a correction in the case of inhomogeneous Dirichlet, Neumann, and mixed boundary conditions are presented. Numerical examples that illustrate the effectiveness and benefits of these corrections are included.

Highly stable multivalue numerical methods

The numerical solution of partial differential equations discretized along the space variables requires the employ of highly stable methods, due to their intrisic multiscale (thus stiff) nature. The purpose of this paper is then the introduction of some building blocks leading to an efficient and accurate treatment of such stiff problems through highly stable multivalue numerical methods. We present a strategy based on a suitable modification of collocation technique which avoids, unlike classical collocation based Runge-Kutta methods, the order reduction phenomenon. Some novel issues on the error analysis, in view of a combined variable stepsize-variable order implementation, are here presented.