TY - JOUR A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Implicit monotone difference methods for scalar conservation laws with source terms JF - Acta Mathematica Vietnamica N2 - In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests. KW - Entropy solution KW - Source term KW - Monotone methods KW - Implicit methods KW - Finite difference methods KW - Conservation laws Y1 - 2020 U6 - https://doi.org/10.1007/s40306-019-00354-1 SN - 2315-4144 N1 - Corresponding author: Andreas Kleefeld VL - 45 SP - 709 EP - 738 PB - Springer Singapore CY - Singapore ER - TY - JOUR A1 - Asante-Asamani, E.O. A1 - Kleefeld, Andreas A1 - Wade, B.A. T1 - A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting JF - Journal of Computational Physics N2 - A second-order L-stable exponential time-differencing (ETD) method is developed by combining an ETD scheme with approximating the matrix exponentials by rational functions having real distinct poles (RDP), together with a dimensional splitting integrating factor technique. A variety of non-linear reaction-diffusion equations in two and three dimensions with either Dirichlet, Neumann, or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second-order implicit-explicit schemes. An additional performance boost is gained through further use of basic parallelization techniques. KW - Exponential time differencing KW - Real distinct pole KW - Dimensional splitting KW - Reaction-diffusion systems KW - Matrix exponential Y1 - 2020 U6 - https://doi.org/10.1016/j.jcp.2020.109490 SN - 0021-9991 N1 - Corresponding author: Andreas Kleefeld VL - 415 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Martín-Vaquero, J. A1 - Kleefeld, Andreas T1 - Solving nonlinear parabolic PDEs in several dimensions: Parallelized ESERK codes JF - Journal of Computational Physics N2 - There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost. Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different. KW - Multi-dimensional partial differential equations KW - Higher-order codes KW - Nonlinear PDEs Y1 - 2020 U6 - https://doi.org/10.1016/j.jcp.2020.109771 SN - 0021-9991 IS - 423 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Kleefeld, Andreas T1 - The hot spots conjecture can be false: some numerical examples JF - Advances in Computational Mathematics N2 - The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well. KW - Numerics KW - Boundary integral equations KW - Potential theory KW - Helmholtz equation KW - Interior Neumann eigenvalues Y1 - 2021 U6 - https://doi.org/10.1007/s10444-021-09911-5 SN - 1019-7168 VL - 47 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Kleefeld, Andreas A1 - Zimmermann, M. ED - Constanda, Christian ED - Bodmann, Bardo E.J. ED - Harris, Paul J. T1 - Computing Elastic Interior Transmission Eigenvalues JF - Integral Methods in Science and Engineering N2 - An alternative method is presented to numerically compute interior elastic transmission eigenvalues for various domains in two dimensions. This is achieved by discretizing the resulting system of boundary integral equations in combination with a nonlinear eigenvalue solver. Numerical results are given to show that this new approach can provide better results than the finite element method when dealing with general domains. Y1 - 2022 SN - 978-3-031-07171-3 U6 - https://doi.org/10.1007/978-3-031-07171-3_10 N1 - Corresponding author: Andreas Kleefeld SP - 139 EP - 155 PB - Birkhäuser CY - Cham ER - TY - JOUR A1 - Chwallek, Constanze A1 - Nawrath, Lara A1 - Krastina, Anzelika A1 - Bruksle, Ieva T1 - Supportive research on sustainable entrepreneurship and business practices JF - SECA Sustainable Entrepreneurship for Climate Action Y1 - 2024 SN - 978-952-316-514-4 (pdf) SN - 2954-1654 (on-line publication) IS - 3 PB - Lapland University of Applied Sciences Ltd CY - Rovaniemi ER - TY - JOUR A1 - Harris, Isaac A1 - Kleefeld, Andreas T1 - Analysis and computation of the transmission eigenvalues with a conductive boundary condition JF - Applicable Analysis N2 - We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber–Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown. KW - Boundary integral equations KW - Inverse spectral problem KW - Conductive boundary condition KW - Transmission eigenvalues Y1 - 2020 U6 - https://doi.org/10.1080/00036811.2020.1789598 SN - 1563-504X VL - 101 IS - 6 SP - 1880 EP - 1895 PB - Taylor & Francis CY - London ER - TY - JOUR A1 - Clausnitzer, Julian A1 - Kleefeld, Andreas T1 - A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C² boundary JF - Discrete and Continuous Dynamical Systems - Series B N2 - We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C² boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach. KW - Nonlinear eigenvalue problems KW - Boundary integral equations, KW - Exponential Euler scheme, KW - Parabolic SPDEs Y1 - 2024 U6 - https://doi.org/10.3934/dcdsb.2023148 SN - 1531-3492 SN - 1553-524X (eISSN) VL - 29 IS - 4 SP - 1624 EP - 1651 PB - AIMS CY - Springfield ER - TY - JOUR A1 - Frauenrath, Tobias A1 - Hezel, Fabian A1 - Renz, Wolfgang A1 - de Geyer d'Orth, Thibaut A1 - Dieringer, Matthias A1 - von Knobelsdorf-Brenkenhoff, Florian A1 - Prothmann, Marcel A1 - Schulz-Menger, Jeanette A1 - Niendorf, Thoralf T1 - Acoustic cardiac triggering: a practical solution for synchronization and gating of cardiovascular magnetic resonance at 7 Tesla JF - Journal of Cardiovascular Magnetic Resonance N2 - Background To demonstrate the applicability of acoustic cardiac triggering (ACT) for imaging of the heart at ultrahigh magnetic fields (7.0 T) by comparing phonocardiogram, conventional vector electrocardiogram (ECG) and traditional pulse oximetry (POX) triggered 2D CINE acquisitions together with (i) a qualitative image quality analysis, (ii) an assessment of the left ventricular function parameter and (iii) an examination of trigger reliability and trigger detection variance derived from the signal waveforms. Results ECG was susceptible to severe distortions at 7.0 T. POX and ACT provided waveforms free of interferences from electromagnetic fields or from magneto-hydrodynamic effects. Frequent R-wave mis-registration occurred in ECG-triggered acquisitions with a failure rate of up to 30% resulting in cardiac motion induced artifacts. ACT and POX triggering produced images free of cardiac motion artefacts. ECG showed a severe jitter in the R-wave detection. POX also showed a trigger jitter of approximately Δt = 72 ms which is equivalent to two cardiac phases. ACT showed a jitter of approximately Δt = 5 ms only. ECG waveforms revealed a standard deviation for the cardiac trigger offset larger than that observed for ACT or POX waveforms. Image quality assessment showed that ACT substantially improved image quality as compared to ECG (image quality score at end-diastole: ECG = 1.7 ± 0.5, ACT = 2.4 ± 0.5, p = 0.04) while the comparison between ECG vs. POX gated acquisitions showed no significant differences in image quality (image quality score: ECG = 1.7 ± 0.5, POX = 2.0 ± 0.5, p = 0.34). Conclusions The applicability of acoustic triggering for cardiac CINE imaging at 7.0 T was demonstrated. ACT's trigger reliability and fidelity are superior to that of ECG and POX. ACT promises to be beneficial for cardiovascular magnetic resonance at ultra-high field strengths including 7.0 T. KW - Interval Time Series KW - Image Quality Score KW - Image Quality Assessment KW - Sound Pressure Level KW - Cardiovascular Magnetic Resonance Y1 - 2010 U6 - https://doi.org/10.1186/1532-429X-12-67 SN - 1532-429X VL - 12 IS - 1 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Kob, Malte A1 - Frauenrath, Tobias T1 - A system for parallel measurement of glottis opening and larynx position JF - Biomedical Signal Processing and Control N2 - The simultaneous assessment of glottal dynamics and larynx position can be beneficial for the diagnosis of disordered voice or speech production and swallowing. Up to now, methods either concentrate on assessment of the glottis opening using optical, acoustical or electrical (electroglottography, EGG) methods, or on visualisation of the larynx position using ultrasound, computer tomography or magnetic resonance imaging techniques. The method presented here makes use of a time-multiplex measurement approach of space-resolved transfer impedances through the larynx. The fast sequence of measurements allows a quasi simultaneous assessment of both larynx position and EGG signal using up to 32 transmit–receive signal paths. The system assesses the dynamic opening status of the glottis as well as the vertical and back/forward motion of the larynx. Two electrode-arrays are used for the measurement of the electrical transfer impedance through the neck in different directions. From the acquired data the global and individual conductivity is calculated as well as a 2D point spatial representation of the minimum impedance. The position information is shown together with classical EGG signals allowing a synchronous visual assessment of glottal area and larynx position. A first application to singing voice analysis is presented that indicate a high potential of the method for use as a non-invasive tool in the diagnosis of voice, speech, and swallowing disorders. KW - EGG KW - Transfer impedance KW - Tomography KW - Larynx position KW - Voice assessment Y1 - 2009 U6 - https://doi.org/10.1016/j.bspc.2009.03.004 SN - 1746-8108 VL - 4 IS - 3 SP - 221 EP - 228 PB - Elsevier CY - Amsterdam ER -