arXiv subjects

Maria López-Fernández

Publications and source records attributed to Maria López-Fernández.

3 recordsLinked to original sources

arXiv2025

Convolution Quadrature for the quasilinear subdiffusion equation

We construct a Convolution Quadrature (CQ) scheme for the quasilinear subdiffusion equation of order $α$ and supply it with the fast and oblivious implementation. In particular, we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the Finite Element Method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. Our estimates are globally optimal for all $0<α<1$ and pointwise for $α\geq 1/2$ in the sense that they reduce to the well-known results for the linear equation. For the semilinear case, our estimates are optimal both globally and locally. As a passing result, we also obtain a discrete Grönwall inequality for the CQ, which is a crucial ingredient in our convergence proof based on the energy method. The paper is concluded with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.

math.NA↗

arXiv2021

Pseudospectral roaming contour integral methods for convection-diffusion equations

We generalize ideas in the recent literature and develop new ones in order to propose a general class of contour integral methods for linear convection-diffusion PDEs and in particular for those arising in finance. These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, López-Fernández and Nino. In summary, the article (i) provides a comparison among three different integration profiles; (ii) proposes a new fast pseudospectral roaming method; (iii) optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant; (iv) focuses extensively on computational aspects and it is the reference of the MATLAB code https://github.com/MattiaManucci/Contour_Integral_Methods.git, where all algorithms described here are implemented.

math.NA↗

arXiv2018

Numerical inverse Laplace transform for convection-diffusion equations

In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is competitive with other contour integral methods available in the literature.

math.NA↗