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    Item type:Publication, Access status: with files ,
    Spectral deferred correction methods for second-order problems
    (2025)
    Akramov, Ikrom
    ;
    Ruprecht, Daniel
    In this thesis, we present a spectral deferred correction (SDC) method for second-order problems. We provide a complete theoretical analysis of the method, demonstrating its convergence and stability for general second-order problems, and we validate our theoretical results with numerical examples. We also introduce a multi-level SDC (MLSDC) method for second-order problems and compare its residual and convergence properties with those of the standard SDC method. Finally, we extend the SDC framework to multiscale problems using the micro–macro MLSDC (M3LSDC) method, which couples fine and coarse models with tailored FAS corrections. In numerical experiments, we show that M3LSDC achieves a smaller residual than both SDC and MLSDC with the same number of iterations.
    Publicationtype: Doctoral Thesis
      50  45
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    Item type:Publication, Access status: without files ,
    Interface layers and coupling conditions on networks for linearized kinetic BGK equation
    (2024-06)
    Akramov, Ikrom
    ;
    Borsche, Raul
    ;
    Eckhard, Nils
    ;
    Klar, Axel
    We consider a linearized kinetic BGK equation and the associated acoustic system on a network. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the solutions of kinetic half-space problems. This work extends the procedure developed in [15], where coupling conditions for a simplified BGK model have been derived. Numerical comparisons between different coupling conditions confirm the accuracy of the proposed approximation.
    Publicationtype: Journal Article
      30
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    Item type:Publication, Access status: without files ,
    Spectral deferred correction methods for second-order problems
    (Society for Industrial & Applied Mathematics (SIAM), 2024-06-01)
    Akramov, Ikrom
    ;
    Götschel, Sebastian
    ;
    Minion, Michael
    ;
    Ruprecht, Daniel
    ;
    Speck, Robert
    Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit, or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nystr\"om method.
    Publicationtype: Journal Article
      50