An almost symmetric Strang splitting scheme for the construction of high order composition methods

In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state $y_{\star}$ into the corresponding nonlinearity $b(y)y$, which results in a linear term $b(y_{\star})y$ whose exact flow can be determined efficiently. Therefore, in the spirit of splitting methods, it is still possible for the numerical simulation to satisfy certain properties of the exact flow. However, Strang splitting is no longer symmetric (even though it is still a second order method) and thus high order composition methods are not easily attainable. We will show that an iterated Strang splitting scheme can be constructed which yields a method that is symmetric up to a given order. This method can then be used to attain high order composition schemes. We will illustrate our theoretical results, up to order six, by conducting numerical experiments for a charged particle in an inhomogeneous electric field, a post-Newtonian computation in celestial mechanics, and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to Strang splitting. For the first example we also perform a comparison with the standard fourth order Runge--Kutta methods and find significant gains in efficiency as well better conservation properties

Exponential Integrators on Graphic Processing Units

In this paper we revisit stencil methods on GPUs in the context of exponential integrators. We further discuss boundary conditions, in the same context, and show that simple boundary conditions (for example, homogeneous Dirichlet or homogeneous Neumann boundary conditions) do not affect the performance if implemented directly into the CUDA kernel. In addition, we show that stencil methods with position-dependent coefficients can be implemented efficiently as well. As an application, we discuss the implementation of exponential integrators for different classes of problems in a single and multi GPU setup (up to 4 GPUs). We further show that for stencil based methods such parallelization can be done very efficiently, while for some unstructured matrices the parallelization to multiple GPUs is severely limited by the throughput of the PCIe bus.Comment: To appear in: Proceedings of the 2013 International Conference on High Performance Computing Simulation (HPCS 2013), IEEE (2013

On the error propagation of semi-Lagrange and Fourier methods for advection problems

In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme.Comment: submitted to Computers & Mathematics with Application

DRGs in Transfusion Medicine and Hemotherapy in Germany

Patients requiring transfusion medicine and hemotherapy in an inpatient setting are incorporated into the German Diagnosis Related Groups (G-DRG) system in multiple ways. Different DRGs exist in Major Diagnostic Category 16 for patients that have been admitted for the treatment of a condition from the field of transfusion medicine. However, the reimbursement might be not cost covering for many cases, and efforts have to be intensified to find adequate definitions and prices. We believe that this can only be successful if health service research is intensified in this field. For patients requiring hemotherapy and transfusion medicine concomitant to the treatment of an underlying disease such as cancer, multiple systems exist to increase remuneration, among them the Patient Clinical Complexity Level (PCCL) and complex constellations to induce DRG splits. For direct reimbursement of high cost products, additional remuneration fees (Zusatzentgelte, ZE) are the most important. In addition, expensive innovations not reflected within the DRGs can be reimbursed after application and negotiation of the New Diagnostic and Treatment Methods (Neue Untersuchungs- und Behandlungsmethoden, NUB) system. The NUB system guarantees that medical progress is put rapidly into clinical practice and prevents financial issues from becoming a stumbling block for the use of innovative drugs and methods