Summation-By-Parts Operators and High-Order Quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. The accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.Comment: 18 pages, 3 figure

A direct procedure for interpolation on a structured curvilinear two-dimensional grid

A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two-dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial derivative information is obtained by finite differences in the computational space. The partial derivatives in physical space are determined by repeated application of the chain rule for partial differentiation. A bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space into unit squares. The interpolation is performed within each unit square using a piecewise bicubic spline

Die virale Gastroenteritis

Zusammenfassung: Viren sind die häufigste Ursache der akuten Gastroenteritis und nicht nur in den Entwicklungsländern ein Problem. Rotaviren bilden bei Säuglingen und Kleinkindern und Noroviren in allen Altersgruppen die Haupterreger. Neben sporadischen Fällen verursachen sie auch bei uns Epidemien in Krankenhäusern, Pflegeheimen, Schulen, Kinderkrippen, Hotels und Restaurants. Während die medizinische Betreuung sporadischer Fälle außer bei Säuglingen kaum Probleme bietet, ist das Management bei Epidemien aufwändig und zeitraubend. Der vorliegende Beitrag ist eine Übersicht über die wichtigsten Erreger, deren Pathogenese, Therapie und die krankenhaushygienische Betreuun

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.Comment: 26 pages, 5 figure

Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations

High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and compressible Navier-Stokes equations are considerably more expensive than the standard divergence-form discretization. In search of a more efficient entropy stable scheme, we extend the entropy-split method for implementation on unstructured grids and investigate its properties. The main ingredients of the scheme are Harten's entropy functions, diagonal-$\mathsf{E}$ summation-by-parts operators with diagonal norm matrix, and entropy conservative simultaneous approximation terms (SATs). We show that the scheme is high-order accurate and entropy conservative on periodic curvilinear unstructured grids for the Euler equations. An entropy stable matrix-type interface dissipation operator is constructed, which can be added to the SATs to obtain an entropy stable semi-discretization. Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta method. Entropy stable viscous SATs, applicable to both the Hadamard-form and entropy-split schemes, are developed for the compressible Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the compressible Navier-Stokes equations. Local conservation in the vicinity of discontinuities is enforced using an entropy stable hybrid scheme. Several numerical problems involving both smooth and discontinuous solutions are investigated to support the theoretical results. Computational cost comparison studies suggest that the entropy-split scheme offers substantial efficiency benefits relative to Hadamard-form multidimensional SBP-SAT discretizations.Comment: 34 pages, 8 figure

Local Epitaxial Overgrowth for Stacked Complementary MOS Transistor Pairs

A three-dimensional silicon processing technology for CMOS circuits was developed and characterized. The first fully depleted SOI devices with individually biasable gates on both sides of the silicon film were realized. A vertically stacked CMOS Inverter built by lateral overgrowth was reported for the first time. Nucleation-free epitaxial lateral overgrowth of silicon over thin oxides was developed for both a pancake and a barrel-type epitaxy reactor: This process was optimized to limit damage to gate oxides and minimize dopant diffusion within the Substrate. Autodoping from impurities of the MOS transistors built in the substrate was greatly reduced. A planarisation technique was developed to reduce the silicon film thickness from 13μm to below 0.5μm for full depletion. Chemo-mechanical polishing was modified to yield an automatic etch stop with the corresponding control and uniformity of the silicon film. The resulting wafer topography is more planar than in a conventional substrate CMOS process. PMOS transistors which match the current drive of bulk NM0S devices of equal geometry were characterized, despite the three-times lower hole mobility. Devices realized in the substrate, at the bottom and on top of the SOI film were essentially indistinguishable from bulk devices. A novel device with two insulated gates controlling the same channel was characterized. Inverters were realized both as joint-gate configuration and with symmetric performance of n- and p-channel. These circuits were realized in the area of a single NMOS transistor