arXiv subjects

Carlos Arranz-Simón

Publications and source records attributed to Carlos Arranz-Simón.

5 recordsLinked to original sources

arXiv2026

Splitting methods for nonlinear Schrödinger equation without order reduction

A technique is provided in this paper to integrate nonlinear Schrödinger equation with time-dependent Dirichlet boundary conditions with high-order Yoshida splittings which are based on Strang method. For that, a modification of Strang method is required in which the linear and stiff part of the equation is integrated with a rational-like version of midpoint rule for which the required boundary values can be calculated without resorting to any differentiation of data. Although Yoshida splitting (with real coefficients) cannot be applied to parabolic problems to obtain order higher than two because of stability, the modified Strang method is also applicable to such type of problems and local order $3$ and global order $2$ are also obtained without differentiation of data.

math.NA↗

arXiv2025

Exponential integrators for parabolic problems with non-homogeneous boundary conditions

Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this paper, we extend classical convergence results to the case of non-homogeneous boundary conditions. Since non-homogeneous boundary conditions typically cause order reduction, we introduce a correction strategy based on smooth extensions of the boundary data. This results in a reformulation as a homogeneous problem with modified source term, to which standard exponential integrators can be applied. For linear problems, we prove that the corrected schemes recover the expected convergence order, and hat higher orders can be attained with suitable quadrature rules, reaching order $2s$ for s-stage Gauss collocation methods. For semilinear problems, our approach preserves the convergence orders guaranteed by exponential Runge-Kutta methods satisfying the corresponding stiff order conditions. Numerical experiments validate the theoretical findings.

math.NA↗

arXiv2025

Rational methods for abstract linear initial boundary value problems without order reduction

Given an $A$-stable rational approximation to $e^z$ of order $p$, numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term $f$ and boundary value $g$. These procedures exhibit the optimal order $p$ and can be implemented by using just one single evaluation of $f$ and $g$ per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for $p\le 6$. The full discretization is also studied and the theoretical results are corroborated by numerical experiments.

math.NA↗

arXiv2025

Rational methods for abstract semilinear problems without order reduction

Rational methods are intended to time integrate linear homogeneous problems. However, their scope can be extended so as to cover linear nonhomogeneous problems. In this paper the integration of semilinear problems is considered. The resulting procedure requires the same computational cost than the one of a linked Runge--Kutta method, with the advantage that the order reduction phenomenon is avoided. Some numerical illustrations are included showing the predicted behaviour of the proposed methods.

math.NA↗

arXiv2024

Rational methods for abstract linear, non-homogeneous problems without order reduction

Starting from an A-stable rational approximation to $\rm{e}^z$ of order $p$, $$r(z)= 1+ z+ \cdots + z^p/ p! + O(z^{p+1}),$$ families of stable methods are proposed to time discretize abstract IVP's of the type $u'(t) = A u(t) + f(t)$. These numerical procedures turn out to be of order $p$, thus overcoming the order reduction phenomenon, and only one evaluation of $f$ per step is required.

math.NA↗