Solvable Test Research Articles

Feline communication strategies when presented with an unsolvable task: the attentional state of the person matters

Research on social cognitive ability in domestic cats is limited. The current study investigated social referencing in cats when exposed to first, a solvable, and then, an unsolvable scenario (i.e., reachable and unreachable treats) in the presence of either an attentive or an inattentive caregiver. Cats expressed more gaze alternation (P = 0.013), but less interaction with the caregiver (P = 0.048) and approached the treat container less frequently (P = 0.017) during the unsolvable test, compared to the solvable test. When in the presence of an attentive caregiver, cats initiated first gaze at the caregiver faster (P = 0.001); gazed at the caregiver for longer (P = 0.034); and approached the treat more frequently (P = 0.040), compared to when the caregiver was inattentive. Significant interaction was observed between test and caregiver’s attentional state on the expression of sequential behavior, a type of showing behavior. Cats exhibited this behavior marginally more with attentive caregivers, compared to inattentive caregivers, but only during the unsolvable test. There was a decrease in sequential behavior during the unsolvable test, compared to solvable test, but this was only seen with inattentive caregivers (P = 0.018). Our results suggest that gaze alternation is a behavior reliably indicating social referencing in cats and that cats’ social communication with humans is affected by the person’s availability for visual interaction.

Improved conductivity and stability of anion exchange membranes by introducing steric hindrance and crosslinked structure

The alternating copolymer based on poly (ether ether ketone) (PEEK) is synthesized with ordered side chain. A series of novel anion exchange membranes grafte with the 1, 2-dimethylimidazole and 1-vinylimidazole are obtained. The copolymer was verified by 1H NMR and the crosslinked membranes are further investigated by solvability test. The ordered hydrophilic side chains form well-defined microphase separation structure, which are proved by Transmission electron micrographs microscopy (TEM). The ionic conductivity is 0.075 S/cm at 80 °C of Im-PEEK-0 uncross-linked membrane. With the addition of 1-vinylimidazole, the maximum stress increases to 66.57 MPa, the water uptake drop to 17.1% and swelling ratio drop to 14.8% at 80 °C of Im-PEEK-0.3 membrane. The hydroxide conductivity remains 82.8% in 2 mol L−1 NaOH solution at 60 °C for 400 h. Meanwhile, all the membranes exhibit excellent thermal stability. Overall, the ordered imidazolium-functionalized side chains provide a method to balance hydroxide conductivity and alkali stability of anion exchange membranes.

Non-coplanar Model States in Quantum Magnetism Applications of the High-Order Coupled Cluster Method

Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases frac{1}{2} le S le 5 . For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Delta ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S le 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.

A complete multi-robot path-planning algorithm

In the domain of multi-robot path-planning problems, robots must move from their start locations to their goal locations while avoiding collisions with each other. The research problem that we addressed is to find a complete solution for the multi-robot path-planning problem. Our first contribution is to recognize the solvable instances of the problem with our solvability test; the theoretical analysis has already been provided to show the validity of this test. Our second contribution is to solve this problem completely, in polynomial time, with the Push and Spin (PASp) algorithm. Once the problem was solved, we found decisions within the complete solution that may improve the performance of the complete algorithm. Hence, our third contribution is to improve the performance by selecting the best path from the set of complete paths. We refer to the improved version of our algorithm as the improved PASp algorithm. In terms of the completeness evaluation, the mathematical proofs demonstrate that the PASp is a complete algorithm for a wider class of problem instances than the classes solved by the Push and Swap (PAS), Push and Rotate (PAR), Bibox or the tractable multi-robot path-planning (MAPP) algorithms. Moreover, PASp solves any graph recognized to be solvable without any assumptions. In addition, the theoretical proof of the PASp algorithm showed completive polynomial performance in terms of total-path-lengths and execution time. In our performance evaluation, the experimental results showed that the PASp performs competitively, in reasonable execution time, in terms of number of moves compared to the PAS, PAR, Bibox and MAPP algorithms on a set of benchmark problems from the video-game industry. In addition, the results showed the scalability and robustness of PASp in problems that can be solved only by PASp. Such problems require high levels of coordination with an efficient number of moves and short execution time. In grid and bi-connected graphs with too many cycles, PASp required more moves and more time than the PAS, PAR and Bibox algorithms. However, PASp is the only algorithm capable of solving such instances with only one unoccupied vertex. Furthermore, adding heuristic search and smooth operation to the improved PASp showed significant further improvement by reducing the number of moves for all problem instances. PASp produced the best plans in a bit higher time. Finally, the PASp algorithm solves a wider class of problems and performs more completely in very complex/crowded environments than other state-of-art algorithms. Additionally, the Spin operation introduces a novel swapping technique to exchange two items and restore others in a graph for industrial applications.

Efficient Implicit Solvation Method for Full Potential DFT.

With the advent of efficient electronic structure methods, effective continuum solvation methods have emerged as a way to, at least partially, include solvent effects into simulations without the need for expensive sampling over solvent degrees of freedom. The multipole moment expansion (MPE) model, while based on ideas initially put forward almost 100 years ago, has recently been updated for the needs of modern electronic structure calculations. Indeed, for an all-electron code relying on localized basis sets and-more importantly-a multipole moment expansion of the electrostatic potential, the MPE method presents a particularly cheap way of solving the macroscopic Poisson equation to determine the electrostatic response of a medium surrounding a solute. In addition to our implementation of the MPE model in the FHI-aims electronic structure theory code [ Blum , V. ; Comput. Phys. Commun. 2009 , 180 , 2175 - 2196 , DOI: 10.1016/j.cpc.2009.06.022 ], we describe novel algorithms for determining equidistributed points on the solvation cavity-defined as a charge density isosurface-and the determination of cavity surface and volume from just this collection of points and their local density gradients. We demonstrate the efficacy of our model on an analytically solvable test case, against high-accuracy finite-element calculations for a set of ≈140000 2D test cases, and finally against experimental solvation free energies of a number of neutral and singly charged molecular test sets [ Andreussi , O. ; J. Chem. Phys. 2012 , 136 , 064102 , DOI: 10.1063/1.3676407 ; Marenich , A. V. ; Minnesota Solvation Database , Version 2012; University of Minnesota : Minneapolis, MN, USA , 2012 . ]. In all test cases we find that our MPE approach compares very well with given references at computational overheads < 20% and sometimes much smaller compared to a plain self-consistency cycle.

A meshless approach to thermomechanics of DC casting of aluminium billets

The ability to model thermomechanics in DC casting is important due to the technological challenges caused by physical phenomena such as different ingot distortions, cracking, hot tearing and residual stress. Many thermomechanical models already exist and usually take into account three contributions: elastic, thermal expansion, and viscoplastic to model the mushy zone. These models are, in a vast majority, solved by the finite element method. In the present work the elastic model that accounts for linear thermal expansion is considered. The method used for solving the model is of a novel meshless type and extends our previous meshless attempts in solving fluid mechanics problems. The solution to the problem is constructed using collocation on the overlapping subdomains, which are composed of computational nodes. Multiquadric radial basis functions, augmented by monomials, are used for the displacement interpolation. The interpolation is constructed in such a manner that it readily satisfies the boundary conditions. The discretization results in construction of a global square sparse matrix representing the system of linear equations for the displacement field. The developed method has many advantages. The system of equations can be easily constructed and efficiently solved. There is no need to perform expensive meshing of the domain and the formulation of the method is similar in two and three dimensions. Since no meshing is required, the nodes can easily be added or removed, which allows for efficient adaption of the node arrangement density. The order of convergence, estimated through an analytically solvable test, can be adjusted through the number of interpolation nodes in the subdomain, with 6 nodes being enough for the second order convergence. Simulations of axisymmetric mechanical problems, associated with low frequency electromagnetic DC casting are presented.

On the Decidability Status of Fuzzy A ℒ C $\mathcal {A}\mathcal {L}\mathcal {C}$ with General Concept Inclusions

The combination of Fuzzy Logics and Description Logics (DLs) has been investigated for at least two decades because such fuzzy DLs can be used to formalize imprecise concepts. In particular, tableau algorithms for crisp Description Logics have been extended to reason also with their fuzzy counterparts. It has turned out, however, that in the presence of general concept inclusion axioms (GCIs) this extension is less straightforward than thought. In fact, a number of tableau algorithms claimed to deal correctly with fuzzy DLs with GCIs have recently been shown to be incorrect. In this paper, we concentrate on fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\), the fuzzy extension of the well-known DL \(\mathcal {A}\mathcal {L}\mathcal {C}\). We present a terminating, sound, and complete tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with arbitrary continuous t-norms. Unfortunately, in the presence of GCIs, this algorithm does not yield a decision procedure for consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies since it uses as a sub-procedure a solvability test for a finitely represented, but possibly infinite, system of inequations over the real interval [0,1], which are built using the t-norm. In general, it is not clear whether this solvability problem is decidable for such infinite systems of inequations. This may depend on the specific t-norm used. In fact, we also show in this paper that consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies with GCIs is undecidable for the product t-norm. This implies, of course, that for the infinite systems of inequations produced by the tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with product t-norm, solvability is in general undecidable. We also give a brief overview of recently obtained (un)decidability results for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) w.r.t. other t-norms.

Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

Spectral parameter power series for perturbed Bessel equations

A spectral parameter power series (SPPS) representation for regular solutions of singular Bessel type Sturm–Liouville equations with complex coefficients is obtained as well as an SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. It is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the Sturm–Liouville problem. Based on the SPPS representation a new mapping property of the transmutation operator for the considered perturbed Bessel operator is obtained, and a new numerical method for solving corresponding spectral problems is developed. The range of applicability of the method includes complex coefficients, complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator. On a set of known test problems we show that the developed numerical method based on the SPPS representation is highly competitive in comparison to the best available solvers such as SLEIGN2, MATSLISE and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the SPPS method delivers excellent numerical results.

Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities

Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated GramCharlier Expansions (GrChE) as possible methods for uncertainty quantification (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents. These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the ‘uncertain’ subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE). Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties in the damping. Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory. Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times. These limitations of the two types of truncated spectral expansions arise from the following: (i) Non-uniform convergence in time of PCE and GrChE resulting in a rapidly increasing number of terms necessary for a good approximation of the random process as time evolves, (ii) Fundamental problems with capturing the constant flux of randomness due to white Gaussian noise forcing via finite truncations of the spectral representation of the associated Wiener process, (iii) Slow decay of PCE and GrChE coefficients in the presence of intermittency, hampering implementation of sparse truncation methods which have been widely used in nearly elliptic problems or in low Reynolds number flows. Rigorous justification of these limitations is richly illustrated by straightforward tests exploiting a simple nonlinear and non-Gaussian but statistically exactly solvable test model which is proposed here as a challenging benchmark for algorithms for UQ in systems with intermittency.

Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.

Challenges in Climate Science and Contemporary Applied Mathematics

AbstractThis article discusses the challenges in climate science from the emerging viewpoint of stochastic‐statistical properties of turbulent dynamical systems. The mathematical topics discussed here include empirical information theory, fluctuation‐dissipation theorems, reduced‐order stochastic modeling, and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems. The applied mathematics topics include the emerging development of multiscale algorithms for filtering/data assimilation and superparametrization for climate science and other problems in science and engineering, as well as suitable unambiguous mathematical test problems for their behavior. Interesting contemporary research directions and specific open problems are mentioned throughout the article. The perspective here should also be useful for applications to other complex dynamical systems arising in neural science, material science, and environmental/mechanical engineering. © 2011 Wiley Periodicals, Inc.

A bi-objective time-dependent vehicle routing and scheduling problem for hazardous materials distribution

Planning hazardous materials distribution routes for servicing a given set of orders within specified time windows is a problem frequently surfacing in a city logistics environment which is characterized by dynamic travel times. The hazardous materials distribution problem involves the determination of the sequence of deliveries and the corresponding paths assigned to each truck. This paper presents the formulation of the hazardous materials distribution problem as a bi-objective time-dependent vehicle routing problem with time windows. The paper presents the mathematical formulation of the problem as an integer network flow model with multiple objectives. The weighted-sum method is applied decomposing the bi-objective vehicle routing and scheduling problem to a series of single-objective instances of the problem, where the objective function is expressed by the weighted sum of the criteria under consideration. A route-building heuristic algorithm is presented for addressing each of the constituent single-objective problems, where stops are inserted iteratively in the front part of the unfinished route. A label-setting algorithm is integrated in the heuristic algorithm for solving the time-dependent shortest path problem with multiple intermediate stops arising after the insertion of any stop in the route. The proposed solution approach has been applied to a set of solvable test problems to assess the accuracy of the heuristic solutions. The results indicate a tolerable deviation of the heuristic solutions from the actual non-dominated solutions. In addition, the proposed algorithm was applied to a set of test problems resembling real-life problem cases. The average computational time needed for solving this type of test problems is not prohibitive.

Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters

The filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model. In particular, real-time filtering and prediction when very limited or no a posteriori analysis is possible (e.g. spread of pollutants, storm surges, tsunami detection, etc.) introduces a number of additional challenges to the problem. Here, a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear, exactly solvable, stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent, transient instabilities associated with positive finite-time Lyapunov exponents to laminar behavior. Stochastic Parameterization Extended Kalman Filter (SPEKF), used as a benchmark here, involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters. The remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and/or linear tangent approximation used for computing the second-order statistics of the augmented stochastic forecast model. A comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases. The estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter, despite its popularity, is completely unreliable in many turbulent regimes for both parameter estimation and filtering; moreover, regimes of filter divergence for the linear tangent filter are identified. The results presented here provide useful guidelines for filtering turbulent, high-dimensional, spatially extended systems with more general model errors, as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi-scale models. They also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended Kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics.

Filtering a statistically exactly solvable test model for turbulent tracers from partial observations

A statistically exactly solvable model for passive tracers is introduced as a test model for the authors’ Nonlinear Extended Kalman Filter (NEKF) as well as other filtering algorithms. The model involves a Gaussian velocity field and a passive tracer governed by the advection–diffusion equation with an imposed mean gradient. The model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere. One of the attractive properties of the model is the existence of the exact statistical solution. In particular, this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real-time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales. Here, we extensively study the performance of the NEKF which uses the exact first and second order nonlinear statistics without any approximations due to linearization. The role of partial and sparse observations, the frequency of observations and the observation noise strength in recovering the true signal, its spectrum, and fat tail probability distribution are the central issues discussed here. The results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations.

Quantifying uncertainty in climate change science through empirical information theory

Quantifying the uncertainty for the present climate and the predictions of climate change in the suite of imperfect Atmosphere Ocean Science (AOS) computer models is a central issue in climate change science. Here, a systematic approach to these issues with firm mathematical underpinning is developed through empirical information theory. An information metric to quantify AOS model errors in the climate is proposed here which incorporates both coarse-grained mean model errors as well as covariance ratios in a transformation invariant fashion. The subtle behavior of model errors with this information metric is quantified in an instructive statistically exactly solvable test model with direct relevance to climate change science including the prototype behavior of tracer gases such as CO(2). Formulas for identifying the most sensitive climate change directions using statistics of the present climate or an AOS model approximation are developed here; these formulas just involve finding the eigenvector associated with the largest eigenvalue of a quadratic form computed through suitable unperturbed climate statistics. These climate change concepts are illustrated on a statistically exactly solvable one-dimensional stochastic model with relevance for low frequency variability of the atmosphere. Viable algorithms for implementation of these concepts are discussed throughout the paper.

Optimal and near–optimal advection–diffusion finite–difference schemes. VI Two–dimensional alternating directions

Computationally efficient alternating–direction implicit finite–difference schemes with high accuracy are derived for the numerical solution on a regular rectangular grid of the two–dimensional constant–coefficient diffusion equation with decay, advection, off–diagonal diffusivity and sources ∂ t c + λ c + u 1 ∂ x 1 c + u 2 ∂ x 2 c - k 11 ∂ x 1 2 c - 2 k 12 ∂ x 1 ∂ x 2 c - k 22 ∂ x 2 2 c = q . The 3×3×2 compact computational module involves three consecutive spatial points in each of the perpendicular spatial directions at two successive time levels. The schemes are tested for three variants of an exactly solvable test case, corresponding to the fractionation of an initial mixture of sinking, neutral and rising particles released together in a vertically well-mixed turbulent open-channel flow with mean current not aligned with either x 1 or x 2.

Post‐Newtonian Smoothed Particle Hydrodynamics

We introduce an adaptation of the well-known tree+SPH numerical scheme to post-Newtonian (PN) hydrodynamics and gravity. Our code solves the (0 + 1 + 2.5)PN equations. These equations include Newtonian hydrodynamics and gravity (0PN), the first-order relativistic corrections to those (1PN), and the lowest order gravitational radiation terms (2.5PN). We test various aspects of our code using analytically solvable test problems. We then proceed to study the 1PN effects on binary neutron star coalescence by comparing calculations with and without the 1PN terms. We find that the effect of the 1PN terms is rather small. The largest effect arises with a stiff equation of state for which the maximum rest mass density increases. This could induce black hole formation. The gravitational wave luminosity is also affected.

Optimal and near-optimal advection-diffusion finite-difference schemes. IV. Spatial non–uniformity

For the non-constant-coefficient advection-diffusion equation with decay a ( ∂ t c + λ c + u ∂ s c ) - ∂ s ( a κ ∂ s c ) = q , the non-uniformity can almost be removed with a local change of variables from c ( s,t ) to b ( x,t ). So, in numerical computations non-uniformity need not imply any reduction from optimal accuracy. A computational grid that was uniformly spaced with respect to s becomes non-uniform with respect to the intrinsic x -coordinate. Suitable non-uniform grid, optimal and near-optimal compact finite-difference schemes were derived in part II of this sequence of papers. The cylindrical advection-diffusion equation is used as an exactly solvable test case.

Relativistic hydrodynamics for heavy-ion collisions. I. General aspects and expansion into vacuum

We present algorithms to solve relativistic hydrodynamics in 3+1--dimensional situations without apparent symmetry to simplify the solution. In simulations of heavy--ion collisions, these numerical schemes have to deal with the physical vacuum and with equations of state with a first order phase transition between hadron matter and a quark--gluon plasma. We investigate their performance for the one--dimensional expansion of baryon-free nuclear matter into the vacuum, which is an analytically solvable test problem that incorporates both the aspect of the vacuum as well as that of a phase transition in the equation of state. The dependence of the lifetime of the mixed phase on the initial energy density is discussed.