Extrapolation applied to certain discretization methods solving the initial value problem for hyperbolic differential equations

Summary The procedurediwiex presented in this paper provides an approximate solution to Cauchy's initial value problem for general hyperbolic systems of first order. The procedurecharex can be applied to the initial value problem for a hyperbolic system of quasi-linear differential equations. This second method is a kind of method of characteristics. It produces a solution for the whole domain of determinancy. Both procedures use extrapolation to the limit. Editor's Note. In this fascile, prepublication of algorithms from the Approximations series of the Handbook for Automatic Computation is continued. Algorithms are published in ALGOL 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones     

Measure space automorphisms,the normalizers of their full groups,and approximate finiteness

We deal with the normalizer $\text{N}$[T] of the full group [T] of a nonsingular transformation T of a Lebesgue measure space in the group of all nonsingular transformations. We solve the conjugacy problem in $\text{N}$[T]/[T] for a measure preserving and ergodic T. Our results show that a locally finite extension of a solvable group is approximately finite.     

Zum Cauchy-Problem bei der verallgemeinerten Wärmeleitungsgleichung

Cauchy's problem for the equationu _xx +x ^–1 u _x =u _t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.     

Effects of boundary conditions on spatially periodic states

We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (“Type-I” solution). A linear stability analysis of the uniform state at threshold reveals that Type-I solutions are often unstable. Then there exists in the full Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (“Type-II” solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.     

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

Synthesis and Crystal Structure of the Strontium Beryllate Sr3Be2O5

Beryllates show an interesting and diverse structural chemistry, resembling that of well-investigated silicates. The coexistence of tetrahedral and trigonal coordination of Be by O atoms in oxoberyllates allows for an even broader variety of structural motives and implies a plurality of possible atomic ratios Be:O in ternary or higher compounds. We have now synthesized the novel strontium oxoberyllate Sr₃Be₂O₅ via a high-temperature high-pressure reaction and have structurally characterized the ternary oxide by single-crystal and powder X-ray diffraction analysis. Sr₃Be₂O₅, a low condensed oxoberyllate, contains unprecedented [Be₂O₅]^6– double triangles and Sr atoms in both double-capped trigonal-prismatic and octahedral coordination. These motifs show striking resemblance to α-SrBeO₂ and SrO, combining their structural properties. Lattice energy (MAPLE) calculations corroborate found parallels to the known phases SrO and α-SrBeO₂.     

Crystalline Nitridophosphates by Ammonothermal Synthesis

Nitridophosphates are a well-studied class of compounds with high structural diversity. However, their synthesis is quite challenging, particularly due to the limited thermal stability of starting materials like P₃N₅. Typically, it requires even high-pressure techniques (e.g. multianvil) in most cases. Herein, we establish the ammonothermal method as a versatile synthetic tool to access nitridophosphates with different degrees of condensation. α-Li₁₀P₄N₁₀, β-Li₁₀P₄N₁₀, Li₁₈P₆N₁₆, Ca₂PN₃, SrP₈N₁₄, and LiPN₂ were synthesized in supercritical NH₃ at temperatures and pressures up to 1070 K and 200 MPa employing ammonobasic conditions. The products were analyzed by powder X-ray diffraction, energy dispersive X-ray spectroscopy, and FTIR spectroscopy. Moreover, we established red phosphorus as a starting material for nitridophosphate synthesis instead of commonly used and not readily available precursors, such as P₃N₅. This opens a promising preparative access to the emerging compound class of nitridophosphates.