arXiv subjects

Mechthild Thalhammer

Publications and source records attributed to Mechthild Thalhammer.

15 recordsLinked to original sources

arXiv2026

Convergence analysis of generalized modified splitting methods using multi-index series

We consider splitting methods for partial differential equations involving unbounded operators. For non-time-reversible dynamics, such as dissipative systems, negative splitting coefficients are generally not admissible because they require stepping backward in time, leading to an order barrier when all coefficients are required to be positive. We introduce generalized modified splitting methods to overcome this barrier. To analyze their convergence, we develop the corresponding multi-index series formalism, which provides a systematic framework for deriving order conditions. Using this formalism, we derive the order conditions and construct a generalized modified splitting method of order $6$. We also provide Python scripts that automate the generation and verification of order conditions, as well as the construction of new generalized modified splitting methods. Finally, we establish connections between the introduced multi-index series and related series formalisms from the literature, in particular, word series, Lie-Butcher series, and multi-index Butcher series.

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arXiv2026

Modified splitting methods for Gross-Pitaevskii systems modelling Bose-Einstein condensates: Time evolution and ground state computation

The year 2025 marks the 100 and 30 years anniversaries of the discovery of Bose--Einstein condensation and its successful experimental realisation. Inspired by these important research achievements, a conceptually simple approach is proposed to facilitate reliable and efficient numerical simulations. The structure of the underlying systems of coupled Gross--Pitaevskii equations suggests the use of optimised high-order operator splitting methods for dynamical evolution and ground state computation. A second-order barrier, however, prevents the applicability of standard operator splitting methods for both, time evolution as well as imaginary time propagation. An innovative alternative approach accomplishes the design of novel modified operator splitting methods that remain stable under moderate smallness assumptions on the time increments. The core idea is to incorporate commutators of the defining differential and nonlinear multiplication operators, since this permits to fulfill the basic stability requirement of positive method coefficients. Further improvements with respect to convergence at the targeted precision arise from automatic adjustments of the time stepsizes by an inexpensive local error control. The presented numerical experiments confirm the favourable performance of a specific fourth-order modified operator splitting method. Amongst others, it is demonstrated that the excellent mass and energy conservation in long-term evolutions, intrinsic attributes of geometric numerical integrators for Hamiltonian systems, is maintained for a sensible variation of the time stepsizes. Moreover, the benefits of adaptive higher-order approximations in ground state computations are illustrated.

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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

Novel approaches for the reliable and efficient numerical evaluation of Landau-type operators

Numerical approximations of Landau-type operators represent fundamental components of time integration methods for demanding problems such as inhomogeneous Vlasov-Landau-type equations. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb-type interaction. This work is concerned with the introduction and numerical comparison of novel approaches for the reliable and efficient evaluation of Landau-type collision operators, where the focus is on the treatment of integral operators involving general singular kernels. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem.

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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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arXiv2022

Community Integration Algorithms (CIAs) for Dynamical Systems on Networks

Dynamics of large-scale network processes underlies crucial phenomena ranging across all sciences. Forward simulation of large network models is often computationally prohibitive. Yet, most networks have intrinsic community structure. We exploit these communities and propose a fast simulation algorithm for network dynamics. In particular, aggregating the inputs a node receives constitutes the limiting factor in numerically simulating large-scale network dynamics. We develop community integration algorithms (CIAs) significantly reducing function-evaluations. We obtain a substantial reduction from polynomial to linear computational complexity. We illustrate our results in multiple applications including classical and higher-order Kuramoto-type systems for synchronisation and Cucker--Smale systems exhibiting flocking behaviour on synthetic as well as real-world networks. Numerical comparison and theoretical analysis confirm the robustness and efficiency of CIAs.

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arXiv2021

On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs

In this work, a novel approach for the reliable and efficient numerical integration of the Kuramoto model on graphs is studied. For this purpose, the notion of order parameters is revisited for the classical Kuramoto model describing all-to-all interactions of a set of oscillators. First numerical experiments confirm that the precomputation of certain sums significantly reduces the computational cost for the evaluation of the right-hand side and hence enables the simulation of high-dimensional systems. In order to design numerical integration methods that are favourable in the context of related dynamical systems on network graphs, the concept of localised order parameters is proposed. In addition, the detection of communities for a complex graph and the transformation of the underlying adjacency matrix to block structure is an essential component for further improvement. It is demonstrated that for a submatrix comprising relatively few coefficients equal to zero, the precomputation of sums is advantageous, whereas straightforward summation is appropriate in the complementary case. Concluding theoretical considerations and numerical comparisons show that the strategy of combining effective community detection algorithms with the localisation of order parameters potentially reduces the computation time by several orders of magnitude.

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arXiv2019

Efficient time integration methods for Gross--Pitaevskii equations with rotation term

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross--Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schr{ö}dinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

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arXiv2019

Splitting and composition methods with embedded error estimators

We propose new local error estimators for splitting and composition methods. They are based on the construction of lower order schemes obtained at each step as a linear combination of the intermediate stages of the integrator, so that the additional computational cost required for their evaluation is almost insignificant. These estimators can be subsequently used to adapt the step size along the integration. Numerical examples show the efficiency of the procedure.

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arXiv2017

Fundamental models in nonlinear acoustics part I. Analytical comparison

This work is concerned with the study of fundamental models from nonlinear acoustics. In Part~I, a hierarchy of nonlinear damped wave equations arising in the description of sound propagation in thermoviscous fluids is deduced. In particular, a rigorous justification of two classical models, the Kuznetsov and Westervelt equations, retained as limiting systems for consistent initial data, is given. Numerical comparisons that confirm and complement the theoretical results are provided in Part~II.

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arXiv2016

Adaptive splitting methods for nonlinear Schr\"{o}dinger equations in the semiclassical regime

The error behavior of exponential operator splitting methods for nonlinear Schr{\"o}dinger equations in the semiclassical regime is studied. For the Lie and Strang splitting methods, the exact form of the local error is determined and the dependence on the semiclassical parameter is identified. This is enabled within a defect-based framework which also suggests asymptotically correct a~posteriori local error estimators as the basis for adaptive time stepsize selection. Numerical examples substantiate and complement the theoretical investigations.

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arXiv2016

Convergence of a Strang splitting finite element discretization for the Schr\"odinger-Poisson equation

Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schr\"odinger equations. In particular, the Schr\"odinger-Poisson equation under homogeneous Dirichlet boundary conditions on a finite domain is considered. A rigorous stability and error analysis is carried out for the second-order Strang splitting method and conforming polynomial finite element discretizations. For sufficiently regular solutions the classical orders of convergence are retained, that is, second-order convergence in time and polynomial convergence in space is proven. The established convergence result is confirmed and complemented by numerical illustrations.

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arXiv2016

The BCH-Formula and Order Conditions for Splitting Methods

As an application of the BCH-formula, order conditions for splitting schemes are derived. The same conditions can be obtained by using non-commutative power series techniques and inspecting the coefficients of Lyndon-Shirshov words.

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arXiv2013

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Efficient time integration methods based on operator splitting are introduced for the Westervelt equation, a nonlinear damped wave equation that arises in nonlinear acoustics as mathematical model for the propagation of sound waves in high intensity ultrasound applications. For the first-order Lie-Trotter splitting method a global error estimate is deduced, confirming that the splitting method remains stable and that the nonstiff convergence order is retained in situations where the problem data are sufficiently regular. Fundamental ingredients in the stability and error analysis are regularity results for the Westervelt equation and related linear evolution equations of hyperbolic and parabolic type. Numerical examples illustrate and complement the theoretical investigations.

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