Energy-conserving methods for the nonlinear Schrödinger equation

In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) (Brugnano et al., 2015), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We shall use HBVMs for solving the nonlinear Schrödinger equation (NLSE), of interest in many applications. We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional, confers more robustness on the numerical solution of such a problem

A conservative numerical method for a time fractional diffusion equation

Geometric numerical integration, the branch of numerical analysis with the goal of finding approximate solutions of differential equations that preserve some structure of the continuous problem, is a well established field of research [5]. In particular, requiring that invariants or conservation laws are preserved, on one hand, applies on the approximations some constraints that are satisfied also by the exact solutions. On the other hand, it guarantees a better propagation of the error over long integration times [3]. In the last two decades, new techniques for finding conservation laws of fractional differential equations have been derived by suitably generalising methods for PDEs [4, 6]. However, the numerical preservation of conservation laws of time fractional differential equations is a research topic still at an embryonic state. This talk deals with the numerical solution of diffusion equations in the form D^α_t u = D^2_x K(u), α ∈ R, where D_x is the partial derivative in space, K is an arbitrary regular function, and D^α_t denotes the Riemann-Liouville fractional derivative of order α. The proposed numerical method combines a finite difference scheme in space with a spectral time integrator and preserves discrete versions of the conservation laws of the original differential equation [1, 2]. The conservative and convergence properties of the proposed method are verified by the computational solution of some numerical experiments. References [1] K. Burrage, A. Cardone, R. D’Ambrosio, B. Paternoster. Numerical solution of time fractional diffusion systems. Appl. Numer. Math., 116 (2017), 82–94. [2] A. Cardone, G. Frasca-Caccia. Numerical conservation laws of time fractional diffusion PDEs. arXiv.2203.01966, (2022). [3] A. Dur´an, J. M. Sanz-Serna. The numerical integration of relative equilibrium solutions. Geometric theory. Nonlinearity, 11, 1547–1567, (1998). [4] G. S. F. Frederico, D. F. M. Torres. Fractional conservation laws in optimal control theory. Nonlinear Dyn., 53 (2008), 215–222. [5] E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, second edition, 2006. [6] S. Y. Lukashchuk. Conservation laws for time-fractional subdiffusion and diffusionwave equations. Nonlinear Dyn., 80 (2015), 791–80

Exponentially fitted methods with a local energy conservation law

A new exponentially fitted version of the discrete variational derivative method for the efficient solution of oscillatory complex Hamiltonian partial differential equations is proposed. When applied to the nonlinear Schrodinger equation, this scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave

On the Computational Solution of a Corrosion Model

We consider a phase field model for studying the corrosion of a 304 stainless steel metal immersed in a sodium chloride solution. Phase field models have been widely used to simulate moving interface problems in a variety of contexts. On one hand, these models have the benefit of treating implicitly the moving boundary by introducing an auxiliary variable. On the other hand, their nature is highly stiff and their computational cost is very high. In this talk we consider convenient numerical techniques for the efficient solution of the considered problem with a focus on their Matlab implementation

Numerical conservation laws of time fractional diffusion PDEs

This paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy discrete counterparts of these conditions have conservation laws that approximate the continuous ones. On the basis of this result, we derive conservation laws for a mixed scheme that combines a finite difference method in space with a spectral integrator in time. A range of numerical experiments shows the convergence of the proposed method and its conservation properties

Line integral solution of Hamiltonian PDEs

In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg-de Vries equation, to illustrate the main features of this novel approach

Hamiltonian boundary value methods (HBVMs) and their efficient implementation

One of the main features when dealing with Hamiltonian problems is the conservation of the energy. In this paper we review, at an elemental level, the main facts concerning the family of low-rank Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) for the efficient numerical integration of these problems. Using these methods one can obtain, an at least “practical”, conservation of the Hamiltonian. We also discuss the efficient implementation of HBVMs by means of two different procedures: the blended implementation of the methods and an iterative procedure based on a particular triangular splitting of the corresponding Butcher’s matrix. We analyze the computational cost of these two procedures that result to be an excellent alternative to a classical fixed-point iteration when the problem at hand is a stiff one. A few numerical tests confirm all the theoretical findings