arXiv subjects

Alexander Ostermann

Publications and source records attributed to Alexander Ostermann.

At least 19 recordsLinked to original sources

arXiv2026

Explicit Fourier Integrator for the Periodic dNLS via Gauge Transformation: Low-Regularity Estimates in Discrete Bourgain Spaces

The derivative nonlinear Schr\"odinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(\tau^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $\tau$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.

math.NA↗

arXiv2026

A low regularity exponential-type integrator for the derivative nonlinear Schr\"odinger equation

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schr\"odinger equation with low regularity. Our analysis shows that for any $s>\frac12$, the method converges with first-order in $H^s(\mathbb{T})$ for initial data $u_0\in H^{s+1}(\mathbb{T})$. Moreover, we constructed a symmetrized version of this method that performs better in terms of both global error and conservation behavior. To the best of our knowledge, these are the first low regularity integrators for the derivative nonlinear Schr\"odinger equation. Numerical experiments illustrate our theoretical findings.

math.NA↗

arXiv2025

A simple predictor-corrector scheme without order reduction for advection-diffusion-reaction problems

Treating diffusion and advection/reaction separately is an effective strategy for solving semilinear advection-diffusion-reaction equations. However, such an approach is prone to suffer from order reduction, especially in the presence of inhomogeneous Dirichlet boundary conditions. In this paper, we extend an approach of Einkemmer and Ostermann [SIAM J. Sci. Comput. 37, A1577-A1592, 2015] to advection-diffusion-reaction problems, where the advection and reaction terms depend nonlinearly on both the solution and its gradient. Starting from a modified splitting method, we construct a predictor-corrector scheme that avoids order reduction and significantly improves accuracy. The predictor only requires the solution of a linear diffusion equation, while the corrector is simply an explicit Euler step of an advection-reaction equation. Under appropriate regularity assumptions on the exact solution, we rigorously establish second-order convergence for this scheme. Numerical experiments are presented to confirm the theoretical results.

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

Exponential Runge-Kutta methods for parabolic equations with state-dependent delay

The aim of this paper is to construct and analyze exponential Runge-Kutta methods for the temporal discretization of a class of semilinear parabolic problems with arbitrary state-dependent delay. First, the well-posedness of the problem is established. Subsequently, first and second order schemes are constructed. They are based on the explicit exponential Runge-Kutta methods, where the delayed solution is approximated by a continuous extension of the time discrete solution. Schemes of arbitrary order can be constructed using the methods of collocation type. The unique solvability and convergence of the proposed schemes are established. Finally, we discuss implementation issues and present some numerical experiments to illustrate our theoretical results.

math.NA↗

arXiv2025

Nonsmooth data error estimates for exponential Runge-Kutta methods and applications to split exponential integrators

We derive error bounds for exponential Runge-Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in a framework of abstract semilinear evolution equations with operators having non-dense domain. In particular, we investigate nonsmooth data error estimates for the Allen-Cahn and the Burgers' equation. As an application, we apply these nonsmooth data error estimates to split exponential integrators and derive a convergence result in terms of the data.

math.NA↗

arXiv2025

An initial-boundary corrected splitting method for diffusion-reaction problems

Strang splitting is a widely used second-order method for solving diffusion-reaction problems. However, its convergence order is often reduced to order $1$ for Dirichlet boundary conditions and to order $1.5$ for Neumann and Robin boundary conditions, leading to lower accuracy and reduced efficiency. In this paper, we consider a new splitting approach, called an initial-boundary corrected splitting, which avoids order reduction while improving computational efficiency for a wider range of applications. In contrast to the corrections proposed in the literature, it does not require the computation of correction terms that depend on the boundary conditions and boundary data. Through rigorous analytical convergence analysis and numerical experiments, we demonstrate the improved accuracy and performance of the proposed method.

math.NA↗

arXiv2025

On the convergence of split exponential integrators for semilinear parabolic problems

Splitting the exponential-like $\varphi$ functions, which typically appear in exponential integrators, is attractive in many situations since it can dramatically reduce the computational cost of the procedure. However, depending on the employed splitting, this can result in order reduction. The aim of this paper is to analyze different such split approximations. We perform the analysis for semilinear problems in the abstract framework of commuting semigroups and derive error bounds that depend, in particular, on whether the vector (to which the $\varphi$ functions are applied) satisfies appropriate boundary conditions. We then present the convergence analysis for two split versions of a second-order exponential Runge--Kutta integrator in the context of analytic semigroups, and show that one suffers from order reduction while the other does not. Numerical results for semidiscretized parabolic PDEs confirm the theoretical findings.

math.NA↗

arXiv2025

Exponential Runge-Kutta methods of collocation type for parabolic equations with time-dependent delay

In this paper, exponential Runge-Kutta methods of collocation type (ERKC) which were originally proposed in (Appl Numer Math 53:323-339, 2005) are extended to semilinear parabolic problems with time-dependent delay. Two classes of the ERKC methods are constructed and their convergence properties are analyzed. It is shown that methods with $s$ arbitrary nonconfluent collocation parameters achieve convergence of order $s$. Provided that the collocation parameters fulfill some additional conditions and the solutions of the problems exhibit sufficient temporal and spatial smoothness, we derive superconvergence results. Finally, some numerical experiments are presented to illustrate our theoretical results.

math.NA↗

arXiv2025

An initial-corrected splitting approach for convection-diffusion-reaction problems

Splitting methods constitute a well-established class of numerical schemes for solving convection-diffusion-reaction problems. They have been shown to be effective in solving problems with periodic boundary conditions. However, in the case of Dirichlet boundary conditions, order reduction has been observed even with homogeneous boundary conditions. In this paper, we propose a novel splitting approach, the so-called `initial-corrected splitting method', which succeeds in overcoming order reduction. A convergence analysis is performed to demonstrate second-order convergence of this modified Strang splitting method. Furthermore, we conduct numerical experiments to illustrate the performance of the newly developed splitting approach.

math.NA↗

arXiv2024

Should exponential integrators be used for advection-dominated problems?

In this paper, we consider the application of exponential integrators to problems that are advection dominated, either on the entire or on a subset of the domain. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators perform comparably to explicit Runge-Kutta schemes for problems that are advection dominated in the entire domain. Moreover, they are able to outperform explicit methods in situations where small parts of the domain are diffusion dominated. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.

math.NA↗

arXiv2024

Highly efficient NURBS-based isogeometric analysis for coupled nonlinear diffusion-reaction equations with and without advection

Nonlinear diffusion-reaction systems model a multitude of physical phenomena. A common situation is biological development modeling where such systems have been widely used to study spatiotemporal phenomena in cell biology. Systems of coupled diffusion-reaction equations are usually subject to some complicated features directly related to their multiphysics nature. Moreover, the presence of advection is source of numerical instabilities, in general, and adds another challenge to these systems. In this study, we propose a NURBS-based isogeometric analysis (IgA) combined with a second-order Strang operator splitting to deal with the multiphysics nature of the problem. The advection part is treated in a semi-Lagrangian framework and the resulting diffusion-reaction equations are then solved using an efficient time-stepping algorithm based on operator splitting. The accuracy of the method is studied by means of a advection-diffusion-reaction system with analytical solution. To further examine the performance of the new method on complex geometries, the well-known Schnakenberg-Turing problem is considered with and without advection. Finally, a Gray-Scott system on a circular domain is also presented. The results obtained demonstrate the efficiency of our new algorithm to accurately reproduce the solution in the presence of complex patterns on complex geometries. Moreover, the new method clarifies the effect of geometry on Turing patterns.

math.NA↗

arXiv2024

The exponential trapezoidal method for semilinear integro-differential equations

The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point iteration, making its implementation straightforward. Second-order convergence in time is shown in an abstract Hilbert space framework under reasonable assumptions on the problem. Numerical experiments illustrate the proven order of convergence.

math.NA↗

arXiv2024

Filtered Lie-Trotter splitting for the "good" Boussinesq equation: low regularity error estimates

We investigate a filtered Lie-Trotter splitting scheme for the ``good" Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in $H^{s}$ for $0 1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order $\tau^{s/2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.

math.NA↗

arXiv2023

Low regularity error estimates for high dimensional nonlinear Schr\"odinger equations

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schr\"{o}dinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in $H^s$ with $0<s\leq 2$. Here, this analysis is extended to dimensions $d=3, 4, 5$ for data satisfying $d/2-1 < s \leq 2$. In this setting, convergence of order $s/2$ in $L^2$ is proven. Numerical examples illustrate these convergence results.

math.NA↗

arXiv2023

Low regularity full error estimates for the cubic nonlinear Schr\"odinger equation

For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(\tau^{s/2}+N^{-s})$ is proved in $L^2$. Here $\tau$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.

math.NA↗

arXiv2023

Accelerating exponential integrators to efficiently solve semilinear advection-diffusion-reaction equations

In this paper we consider an approach to improve the performance of exponential Runge--Kutta integrators and Lawson schemes} in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g. by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and {\color{black} extensive} numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge--Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions.

math.NA↗

arXiv2023

A robust and conservative dynamical low-rank algorithm

Dynamical low-rank approximation, as has been demonstrated recently, can be extremely efficient in solving kinetic equations. However, a major deficiency is that they do not preserve the structure of the underlying physical problem. For example, the classic dynamical low-rank methods violate mass, momentum, and energy conservation. In [L. Einkemmer, I. Joseph, J. Comput. Phys. 443:110495, 2021] a conservative dynamical low-rank approach has been proposed. However, directly integrating the resulting equations of motion, similar to the classic dynamical low-rank approach, results in an ill-posed scheme. In this work we propose a robust, i.e. well-posed, integrator for the conservative dynamical low-rank approach that conserves mass and momentum (up to machine precision) and significantly improves energy conservation. We also report improved qualitative results for some problems and show how the approach can be combined with a rank adaptive scheme.

math.NA↗