Diego Rial

Roberto Ben, Agustín Besteiro, Diego Rial

We propose a high-order numerical methodology for computing the ground state and time evolution of the two-dimensional Gross-Pitaevskii equation with harmonic trapping potential. The ground state is obtained by combining normalized gradient flow with time-splitting schemes featuring strictly positive coefficients, which makes them suitable for both dissipative and Hamiltonian regimes and avoids the negative time steps required by classical symplectic methods. The computational cost of these methods grows quadratically with the order, in contrast to the exponential growth of symplectic alternatives. Numerical benchmarks assess ground state convergence under different initializations, long-time preservation of mass and Hamiltonian energy for varying mass constraints, and the cost-accuracy trade-off for orders q = 2, 4,..., 14.

Agustín Besteiro, Diego Rial

In this paper, we study the initial value problem for infinite dimensional fractional non-autonomous reaction-diffusion equations. Applying general time-splitting methods, we prove the existence of solutions globally defined in time using convex sets as invariant regions. We expose examples, where biological and pattern formation systems, under suitable assumptions, achieve global existence. We also analyze the asymptotic behavior of solutions.

Juan Pablo Borgna, Mariano De Leo, Diego Rial et al.

In this paper we present a unified picture concerning Lie-Trotter method for solving a large class of semilinear problems: nonlinear Schrödinger, Schröginger--Poisson, Gross--Pitaevskii, etc. This picture includes more general schemes such as Strang and Ruth--Yoshida. The convergence result is presented in suitable Hilbert spaces related with the time regularity of the solution and is based on Lipschitz estimates for the nonlinearity. In addition, with extra requirements both on the regularity of the initial datum and on the nonlinearity we show the linear convergence of the method.

Mariano De Leo, Diego Rial, Constanza Sanchez de la Vega

In this work, we show high order splitting methods of integration without negative steps, allowing us to solve numerically irreversible problems, like reaction-diffusion equations. The methods consist in a suitable affine combinations of Lie-Trotter schemes with different steps. We prove convergence of this methods for a large class of semi-linear problems, that includes Hamiltonian and reaction-diffusion systems.