arXiv subjects

Soledad Fernández-García

Publications and source records attributed to Soledad Fernández-García.

6 recordsLinked to original sources

arXiv2025

Multi-rate Runge-Kutta methods: stability analysis and applications

We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we introduce a novel stability analysis, that covers the multi-rate extensions of all standard Runge-Kutta methods and allows to assess the impact of different interpolation methods for the latent variables and of the use of an arbitrary number of sub-steps for the active variables. The stability analysis applies successfully to the model problem typically used in the literature for multi-rate methods. Furthermore,} we also propose a physically motivated model problem that can be used to assess stability to problems with purely imaginary eigenvalues and in situations closer to those arising in applications. Finally, we present an efficient implementation of multi-rate Runge-Kutta methods in the framework of the OpenModelica open-source modelling and simulation software. Results of several numerical experiments, performed with this implementation of the proposed methods, demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.

math.NA↗

arXiv2025

Efficient implicit solvers for models of neuronal networks

We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. These reformulations allow to achieve a significant increase in the efficiency of network simulations, by reducing the size of the algebraic systems effectively solved at each time step. While we focus here specifically on Explicit first step, Diagonally Implicit Runge Kutta methods (ESDIRK), similar simplifications can also be applied to any implicit ODE solver. In order to demonstrate the capabilities of the proposed methods, we consider networks based on three different single-cell models with slow-fast dynamics, including the classical FitzHugh-Nagumo model, a Intracellular Calcium Concentration model and the Hindmarsh-Rose model. Numerical experiments on the simulation of networks of increasing size based on these models demonstrate the superior efficiency of the proposed economical methods.

math.NA↗

arXiv2022

Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations

In this work, we introduce a Variational Multi-Scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements. We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity. We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods.

math.NA↗

arXiv2020

Saddle-node canard cycles in planar piecewise linear differential systems

By applying a singular perturbation approach, canard limit cycles exhibited by a general family of singularly perturbed planar piecewise linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both a supercritical and a subcritical Hopf bifurcation. The obtained results are completely comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Krupa and Szmolyan [18]. In addition, some novel slow-fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle-node bifurcation of canard limit cycles, one involving headless canard cycles, whereas the other involving canard cycles with head. This configuration allows the coexistence of three canard limit cycles.

math.DS↗

arXiv2018

VMS spectral solution of two-dimensional advection-diffusion problems with anisotropic velocity

In this article, we extend the Variational Multi-scale method with spectral approximation of the sub-scales to two-dimensional advection-diffusion problems. The spectral VMS method is cast for low-order elements as a standard VMS method with specific stabilized coefficients associated to a component of the advection velocity. We compute the stabilized coefficients for grids of isosceles right triangles and right quadrilaterals, based upon the explicit computation of the eigen-pairs of the advection-diffusion operator with Dirichlet boundary conditions. To reduce the computing time, the stabilized coefficients are computed at the nodes of a grid in an off-line step, and then interpolated by a fast procedure in the on-line computation. Finally, we present some numerical tests, first with constant velocity and after that with anisotropic variable velocity, in order to compare our results with those provided by other stabilization coefficients. We observe a relevant accuracy gain for moderately large grid Péclet numbers for variable advection velocity.

math.NA↗

arXiv2018

Variational multi-scale spectral solution of convection-dominated parabolic problems

In this work, we consider an extension to parabolic problems of the variational multiscale method with spectral approximation of the sub-scales. We first discretize in time using a finite difference scheme and second, apply the generalization of the spectral variational multi-scale method. To obtain error estimations in convection-dominated flows, we find a helpful link between the stabilized term expressed in terms of Green's functions and in terms of spectral functions. Finally, we present some numerical tests to show the reliability of the method. We consider the stationary one-dimensional advection-diffusion-reaction equation and the evolutive one-dimensional advection-diffusion equation.

math.NA↗