arXiv subjects

Fernando Casas

Publications and source records attributed to Fernando Casas.

At least 19 recordsLinked to original sources

arXiv2026

On a class of modified Cayley--Magnus methods

We introduce a new class of numerical integrators for the time integration of non-autonomous linear ordinary differential equations whose coefficient matrix is sparse and evolves within a quadratic matrix Lie group. In contrast to standard Lie group integrators, the proposed methods avoid the evaluation of matrix exponentials acting on vectors and instead rely on solving a sequence of linear systems with sparse coefficient matrices. Moreover, they are well suited for problems arising from unbounded operators, as they inherently produce bounded solutions. We construct optimised schemes of orders four and six and assess their performance on a representative numerical example, demonstrating clear advantages over existing Lie-group integrators.

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arXiv2026

Error bounds for splitting methods in unitary problems

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.

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arXiv2024

Splitting methods with complex coefficients for linear and nonlinear evolution equations

This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The standard class of splitting methods involving real coefficients is contrasted with an alternative approach that relies on the incorporation of complex coefficients. In view of long-term computations for linear evolution equations, it is expedient to distinguish symmetric, symmetric-conjugate, and alternating-conjugate schemes. The scope of applications comprises high-order reaction-diffusion equations and complex Ginzburg-Landau equations, which are of relevance in the theories of patterns and superconductivity. Time-dependent Gross-Pitaevskii equations and their parabolic counterparts, which model the dynamics of Bose-Einstein condensates and arise in ground state computations, are formally included as special cases. Numerical experiments confirm the validity of theoretical stability conditions and global error bounds as well as the benefits of higher-order complex splitting methods in comparison with standard schemes.

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arXiv2024

Splitting Methods for differential equations

This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyze in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

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arXiv2024

Generalized extrapolation methods based on compositions of a basic 2nd-order scheme

We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization of extrapolation methods and multi-product expansions. A general analysis is provided and new methods up to order 8 are built and tested. The new approach is shown to reduce the latency problem when implemented in a parallel environment and leads to schemes that are significantly more efficient than standard extrapolation when the linear combination is delayed by a number of steps.

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arXiv2024

Families of efficient low order processed composition methods

New families of composition methods with processing of order 4 and 6 are presented and analyzed. They are specifically designed to be used for the numerical integration of differential equations whose vector field is separated into three or more parts which are explicitly solvable. The new schemes are shown to be more efficient than previous state-of-the-art splitting methods.

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arXiv2024

A unifying framework for perturbative exponential factorizations

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme intermediate expansions can also be envisaged. Recurrence formulas are developed. A new lower bound for the convergence of the Wilcox expansion is provided as well as some applications of the results. In particular, two examples are worked out up to high order of approximation to illustrate the behavior of the Wilcox expansion.

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arXiv2024

Exponential perturbative expansions and coordinate transformations

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet--Magnus expansion for periodic systems, the quantum averaging technique and the Lie--Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.

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arXiv2024

Symmetric-conjugate splitting methods for evolution equations of parabolic type

The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schrödinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs.

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arXiv2023

Generalization of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schrödinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schr{ö}dinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schr{ö}dinger case.} Numerical illustrations for the time-dependent Gross--Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods. The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross--Pitaevskii systems in real and imaginary time.

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arXiv2023

Symmetric-conjugate splitting methods for linear unitary problems

We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.

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arXiv2023

A New Optimality Property of Strang's Splitting

For systems of the form $\dot q = M^{-1} p$, $\dot p = -Aq+f(q)$, common in many applications, we analyze splitting integrators based on the (linear/nonlinear) split systems $\dot q = M^{-1} p$, $\dot p = -Aq$ and $\dot q = 0$, $\dot p = f(q)$. We show that the well-known Strang splitting is optimally stable in the sense that, when applied to a relevant model problem, it has a larger stability region than alternative integrators. This generalizes a well-known property of the common Störmer/Verlet/leapfrog algorithm, which of course arises from Strang splitting based on the (kinetic/potential) split systems $\dot q = M^{-1} p$, $\dot p = 0$ and $\dot q = 0$, $\dot p = -Aq+f(q)$.

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arXiv2022

Magnus integrators for linear and quasilinear delay differential equations

A procedure to numerically integrate non-autonomous linear delay differential equations is presented. It is based on the use of an spectral discretization of the delayed part to transform the original problem into a matrix linear ordinary differential equation which is subsequently solved with numerical integrators obtained from the Magnus expansion. The algorithm can be used in the periodic case to get both accurate approximations of the characteristic multipliers and the solution itself. In addition, it can be extended to deal with certain quasilinear delay equations.

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arXiv2022

Split Hamiltonian Monte Carlo revisited

We study Hamiltonian Monte Carlo (HMC) samplers based on splitting the Hamiltonian $H$ as $H_0(θ,p)+U_1(θ)$, where $H_0$ is quadratic and $U_1$ small. We show that, in general, such samplers suffer from stepsize stability restrictions similar to those of algorithms based on the standard leapfrog integrator. The restrictions may be circumvented by preconditioning the dynamics. Numerical experiments show that, when the $H_0(θ,p)+U_1(θ)$ splitting is combined with preconditioning, it is possible to construct samplers far more efficient than standard leapfrog HMC.

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arXiv2021

An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation

We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix-matrix products, respectively. For problems of the form $\exp(-iA)$, with $A$ a real and symmetric matrix, an improved version is presented that computes the sine and cosine of $A$ with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Padé approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schrödinger equation.

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arXiv2021

High order integrators obtained by linear combinations of symmetric-conjugate compositions

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic time-symmetric integrator of order 2n (n $\ge$ 1). The new integrators are of order 2(n + k), k = 1, 2, ..., and preserve time-symmetry up to order 4n + 3 when applied to differential equations with real vector fields. If in addition the system is Hamiltonian and the basic scheme is symplectic, then they also preserve symplecticity up to order 4n + 3. We show that these integrators are well suited for a parallel implementation, thus improving their efficiency. Methods up to order 10 based on a 4th-order integrator are built and tested in comparison with other standard procedures to increase the order of a basic scheme.

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arXiv2021

On symmetric-conjugate composition methods in the numerical integration of differential equations

We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.

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arXiv2020

Computing the matrix sine and cosine simultaneously with a reduced number of products

A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be used in single and double precision arithmetic. The resulting algorithms are more efficient than schemes based on Padé approximations for a wide range of norm matrices.

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