arXiv subjects

Malik Scheifinger

Publications and source records attributed to Malik Scheifinger.

3 recordsLinked to original sources

arXiv2026

Efficient higher-order local time integration for Friedrichs' systems

In this paper, we construct an efficient higher-order local time integration scheme for spatially discretized linear Friedrichs' systems. In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. The special combination of two methods like the leapfrog method on the coarse part of the mesh and the Crank-Nicolson method on the fine part as was done in Hochbruck, Sturm 2016 and Hochbruck, K\"ohler 2022 is not suitable for higher-order time integration. Therefore, we suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal 1991. The techniques developed here for linear problems also carry over to nonlinear problems, where linear systems of the same type arise within a Newton-type iteration. Motivated by the analysis of locally implicit methods by Hochbruck and Sturm 2016, we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory. The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient. We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauss-Legendre Runge--Kutta method.

math.NA↗

arXiv2025

Time-integration of Gaussian variational approximation for the magnetic Schr\"odinger equation

In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schr\"odinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.

math.NA↗

arXiv2025

Local time-integration for Friedrichs' systems

In this paper, we address the full discretization of Friedrichs' systems with a two-field structure, such as Maxwell's equations or the acoustic wave equation in div-grad form, cf. [14]. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh. When using explicit time-integration schemes, the time step size is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme which is motivated by [5]. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local-time stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method, proposed and analyzed in [24]. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.

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