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The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.

Solute transport is usually modeled by the advection-dispersion-reaction equation. In the standard approach, mechanical dispersion is a tensor with principal directions parallel and perpendicular to the flow vector. Since realistic scenarios include nonuniform and unsteady flow fields, the governing equation has full tensor mechanical dispersion. When conventional grid-based numerical methods are used, approximation of the cross terms arising from the off-diagonal terms cause nonphysical solution with oscillations. As an example, for the common scenario of contaminant input into a domain with zero initial concentration, the cross-dispersion terms can result in negative concentrations that can wreak havoc in reactive transport applications. To address this issue, we use the well-known flux-corrected-transport (FCT) technique for a standard finite volume method. Although FCT has most often been used to eliminate oscillations resulting from discretization of the advection term for explicit time stepping, we show that it can be adapted for full-tensor dispersion and implicit time stepping. Unlike other approaches based on new discretization techniques (e.g., mimetic finite difference, nonlinear finite volume), FCT has the advantage of being flexible and widely applicable. Implementation of FCT requires solving an additional system of equations at each time step, using a modified "low order" matrix and a modified right-hand-side vector. To demonstrate the flexibility of FCT, we have modified the well-known and widely used groundwater solute transport simulator, MT3DMS. We apply the new simulator, MT3DMS-FCT, to several benchmark problems that suffer from negative concentrations when using MT3DMS. The new results are mass conservative and strictly nonnegative.

In this work, the enumeration algorithms presented in parts I and II for the globally optimal synthesis of heat exchanger networks are extended to consider non-isothermal mixing. The previous models are modified by adding non-isothermal mixing constraints and new models are constructed to target the bounds of the energy consumption and the binding exchanger minimum approximation temperature. These new models are solved using algorithms that involve solving the solution of systems of equations instead of mathematical programming. We also present two alternatives for optimizing each enumerated structure, namely, the use of a global solver, or the use of a golden search with simple resolution of non-isothermal mixing model for fixed energy consumption. The non-isothermal mixing model is reformulated as a convex model, either solved using nonlinear programming or a programming-free methodology, i.e. solving Karush-Kuhn-Tucker equations. A global optimum search algorithm is developed and examples are tested comparing the proposed strategies.

This chapter develops a reciprocity‐based time domain integral equation technique for analyzing pulsed electromagnetic (EM) scattering from a straight thin‐wire segment. It employs the reciprocity theorem of the time‐convolution type and formulates the antenna problem via the interaction between the (actual) scattered field and the (computational) testing field. The resulting interaction quantity is subsequently represented using a wave‐slowness representation taken along the wire's axis. The chapter shows that for appropriate expansion functions, the transform‐domain interaction quantity can be evaluated analytically in closed form via the Cagniard‐DeHoop technique. It provides a system of equations whose solution is attainable in an updating, step‐by‐step manner. The resulting computational procedure is referred to as the Cagniard‐DeHoop Method of Moments. The chapter also analyzes the excitation of the wire antenna by a uniform EM plane wave and by a voltage delta gap source.

We present a concept of the hybrid finite volume–integral equation technique for solving Maxwell's equation in a quasi‐static form. The divergence correction was incorporated to improve the convergence and stability of the governing linear system equations which pose a challenge on the discretization of the curl–curl Helmholtz equation. A staggered finite volume approach is applied for discretizing the system of equations on a structured mesh and solved in a secondary field technique. The bi‐conjugate gradient stabilizer was utilized with block incomplete lower‐upper factorization preconditioner to solve the system of equation. To obtain the electric and magnetic fields at the receivers, we use the integral Green tensor scheme. We verify the strength of our hybrid technique with benchmark models relative to other numerical algorithms. Importantly, from the tested models, our scheme was in close agreement with the semi‐analytical solution. It also revealed that the use of a quasi‐analytical boundary condition helps to minimize the runtime for the linear system equation. Furthermore, the integral Green tensor approach to compute at the receivers demonstrates better accuracy compared with the conventional interpolation method. This adopted technique can be applied efficiently to the inversion procedure.

Recent work by the author on mixed frequency data analysis has focused on the estimation of cointegrated systems in continuous time based on a fully specified dynamic system of equations, while the estimation of cointegrating vectors in a discrete time system has been approached using a semiparametric frequency domain estimator. We extend the latter approach to cover the continuous time case, establishing the asymptotic properties of the frequency domain estimator and explore, in a simulation study, the effects of misspecifying the continuous time dynamic model in discrete time compared to treating the dynamics non‐parametrically. An empirical illustration is also provided.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables., Comment: 19 pages, 1 figure; July 7, 2020: an addendum is included at the end of the text to fill a gap in the proof of Theorem 1.11. This patch does not change the statement of Theorem 1.11 and other parts of the paper

Author(s): Diaz, M; Boos, WR | Abstract: Although monsoon depressions are a principal synoptic-scale element of the South Asian monsoon, producing extreme rainfall over India and surrounding regions, there exists no widely accepted mechanism explaining their occurrence. This study presents a hierarchy of numerical experiments aimed at finding such an explanation. Using a perturbation-basic state decomposition, we derive an anelastic system of equations that can represent disturbances growing in the complex, three-dimensional monsoon basic state. We find that modal solutions to these equations linearized about this basic state can explain many features of observed monsoon depressions, including their warm-over-cold core structure, westward propagation, and lower-tropospheric wind maximum. For the zonally symmetric case, these modes are barotropically unstable, drawing energy from the meridional shear of the monsoon trough. For the zonally varying basic state, modal solutions still derive energy from barotropic conversion, but fail to achieve positive net growth rates when dissipative processes are included. For the nonlinear equation set, these modes can be excited by a heating impulse, and their energy then remains roughly constant over several days as barotropic energy transfers oppose dissipative losses. Our results support the idea that the general concept of barotropic instability can explain the structure, propagation, and geographic distribution of monsoon depressions, but not their rapid growth rates. We speculate that condensational heating coupled to these waves is needed to obtain a positive growth rate.

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply whilst also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimisation problems, before discussing preconditioners constructed with less standard objectives in mind.

This paper introduces a novel method of modelling acoustic and elastic wave propagation in inhomogeneous media with sharp variations of physical properties based on the recently developed grid-characteristic method which considers different types of waves generated in inhomogeneous linear-elastic media (e.g., longitudinal, transverse, Stoneley, Rayleigh, scattered PP-, SS-waves, and converted PS- and SP-waves). In the framework of this method, the problem of solving acoustic or elastic wave equations is reduced to the interpolation of the solutions, determined at earlier time, thus avoiding a direct solution of the large systems of linear equations required by the FD or FE methods. We apply the grid-characteristic method to compare wave phenomena computed using the acoustic and elastic wave equations in geological medium containing a hydrocarbon reservoir or a fracture zone. The results of this study demonstrate that the developed algorithm can be used as an effective technique for modelling wave phenomena in the models containing hydrocarbon reservoir and/or the fracture zones, which are important targets of seismic exploration.

Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment.

A new method for the estimation of a vector moving average (VMA) process is presented. The technique uses Kullback–Leibler discrepancy with inverse spectra, and yields a Yule–Walker system of equations in inverse autocovariances for the VMA coefficients. This provides a direct formula for the coefficients, which always results in a stable matrix polynomial. The paper provides asymptotic results, as well as an analysis of the method's performance, in terms of speed, bias, and precision. Applications to preliminary estimation of VMA models are discussed, and the method is illustrated on retail data.

SUMMARY In this paper, a new numerical method, Element Differential Method (EDM), is proposed for solving general thermal-mechanical problems. The key point of the method is the direct differentiation of the shape functions of Lagrange isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first and second order partial derivatives of the shape functions with respect to global coordinates are derived. Based on these expressions, a new collocation method is proposed for establishing the system of equations, in which the equilibrium equations are collocated at nodes inside elements, and the traction equilibrium equations are collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements which can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main features of EDM are that no mathematical or mechanical principles are required to set up the system of equations and no integrals are involved to form the coefficients of the system. A number of numerical examples of two- and three-dimensional problems are given to demonstrate the correctness and efficiency of the proposed method.

Inversion of the scalar and vector ray transforms is performed in domain R3\B3(0,r0), ie, with the presence of an obstacle or singularity in the origin. Initially, the ray transforms of the basis functions for the scalar and vector fields are evaluated in an analytical form, and next, the inversion procedure is reduced to a linear system of equations by the use of the least squares method.

Multi-parametric programming has proven to be an invaluable tool for optimisation under uncertainty. Despite the theoretical developments in this area, the ability to handle uncertain parameters on the left-hand side remains limited and as a result, hybrid, or approximate solution strategies have been proposed in the literature. In this work, a new algorithm is introduced for the exact solution of multi-parametric linear programming problems with simultaneous variations in the objective function's coefficients, the right-hand side and the left-hand side of the constraints. The proposed methodology is based on the analytical solution of the system of equations derived from the first order Karush–Kuhn–Tucker conditions for general linear programming problems using symbolic manipulation. Emphasis is given on the ability of the proposed methodology to handle efficiently the LHS uncertainty by computing exactly the corresponding nonconvex critical regions while numerical studies underline further the advantages of the proposed methodology, when compared to existing algorithms.

Directly from the second order differential equations of satellite motion, the linearized orbital perturbation differential equations for CHAMP-like satellites are derived after introducing the reference orbit, and then introducing the omitted terms into the linearized orbital perturbation differential equations, the orbital perturbation differential equations with nonlinear corrections are derived. The accuracies for the orbital perturbation differential equations are estimated and the following results are obtained: if the measurement errors of the satellite positions and the non-gravitational accelerations are less than 3 cm and 3 × 10−10 m·s−2 respectively, the linearized orbital perturbation differential equations and the equations with nonlinear corrections can hold the accuracies 3 × 10−10 m·s−2 only when ρ ≤ 4.7 m and ρ ≤ 4.14 × 103 m respectively, where ρ is the distance between the satellite orbit and the reference one. Hence, compared with the linearized orbital perturbation differential equations, the equations with nonlinear corrections are suitable to establish normal system of equations of the gravity field's spherical harmonic coefficients in long time span. The solving method for the orbital perturbation differential equations is also given with the help of the superposition principle in the paper. At last, some imitation examples for CHAMP and GRACE missions are computed, and the results illustrate that the orbital perturbation differential equations with nonlinear corrections have higher accuracies than the linearized ones.

We consider the problem of small oscillations of a rotating inviscid incompressible fluid. From a mathematical point of view, new exact solutions to the two-dimensional Poincare–Sobolev equation in a class of domains including trapezoid are found in an explicit form and their main properties are described. These solutions correspond to the absolutely continuous spectrum of a linear operator that is associated with this system of equations. For specialists in Astrophysics and Geophysics, the existence of these solutions signifies the existence of some previously unknown type of inertial waves corresponding to the continuous spectrum of inertial oscillations. A fundamental distinction between monochromatic inertial waves and waves of the new type is shown: usual characteristics (frequency, amplitude, wave vector, dispersion relation, direction of energy propagation, and so on) are not applicable to the last. Main properties of these waves are described. In particular, it is proved that they are progressive. Main features of their energy transfer are described.

A Matlab workbook capable of generating complete solutions to a set of linear equations is presented. While standard routines in Matlab and Maple provide solutions to a set of equations, additional work is needed to interpret and relate them to the concepts covered in an undergraduate course in Linear Algebra. One also runs into difficulties when the system of equations is inconsistent. The workbook, started as a simple demo for students, evolved into its present form where the solutions to a set of linear equations are displayed on a single page with explanations, annotations, and equations that match the work by the instructor in class. Regardless of whether the system is consistent or inconsistent with general or unique solutions, the solution set is presented with information such as the rank of a matrix, pivots, null space, free and basic variables, etc., providing the pedagogy. The solutions are compared to those obtained using standard packages from Matlab and Maple demonstrating their shortcomings and limits. In contrast, the results of the workbook provide complete instructive aspects of solutions to a set of linear equations enhancing the educational experience of students. © 2017 Wiley Periodicals, Inc. Comput Appl Eng Educ 25:345–351, 2017; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21803

We consider a mixed finite-volume finite-element method applied to the Navier–Stokes system of equations describing the motion of a compressible, barotropic, viscous fluid. We show convergence as well as error estimates for the family of numerical solutions on condition that: (a) the underlying physical domain as well as the data are smooth; (b) the time step Δ t and the parameter h of the spatial discretization are proportional, Δ t ≈ h ; and (c) the family of numerical densities remains bounded for Δ t , h → 0 . No a priori smoothness is required for the limit (exact) solution. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1208–1223, 2017

Mathematical modeling of thrombosis typically involves modeling the coagulation cascade. Models of coagulation generally involve the reaction kinetics for dozens of proteins. The resulting system of equations is difficult to parameterize, and its numerical solution is challenging when coupled to blood flow or other physics important to clotting. Prior research suggests that essential aspects of coagulation may be reproduced by simpler models. This evidence motivates a systematic approach to model reduction. We herein introduce an automated framework to generate reduced-order models of blood coagulation. The framework consists of nested optimizations, where an outer optimization selects the optimal species for the reduced-order model and an inner optimization selects the optimal reaction rates for the new coagulation network. The framework was tested on an established 34-species coagulation model to rigorously consider what level of model fidelity is necessary to capture essential coagulation dynamics. The results indicate that a nine-species reduced-order model is sufficient to reproduce the thrombin dynamics of the benchmark 34-species model for a range of tissue factor concentrations, including those not included in the optimization process. Further model reduction begins to compromise the ability to capture the thrombin generation process. The framework proposed herein enables automated development of reduced-order models of coagulation that maintain essential dynamics used to model thrombosis.

We propose a formulation for linear elastic fracture mechanics (LEFM) in which the stress intensity factors (SIF) are found directly from the solution vector of an extended boundary element method (XBEM) formulation. The enrichment is embedded in the BEM formulation, rather than adding new degrees of freedom for each enriched node. Therefore, a very limited number of new degrees of freedom is added to the problem, which contributes to preserving the conditioning of the linear system of equations. The Stroh formalism is used to provide BEM fundamental solutions for any degree of anisotropy, and these are used for both conventional and enriched degrees of freedom. Several numerical examples are shown with benchmark solutions to validate the proposed method.

The numerical computation of multicomponent mixture critical points has been the subject of much study due to their theoretical and practical importance. Both deterministic and stochastic methods have been applied with varying degrees of reliability and robustness. In this work, we utilize numerical polynomial homotopy continuation (NPHC) to reliably identify all mixture critical points. This method is unique due to its robustness, initialization-free nature and ease of parallelization. For a given system of equations, all complex solutions are found. Computational times are also found to be invariant to mixture composition. We validate this technique against previous work and extend the method to mixtures of up to eight components. NPHC is shown to be a modern and powerful technique which offers mathematical reliability at a moderate computational cost. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4497–4507, 2016

A new formulation of the classical Navier-Stokes equations is presented, which overcomes the equations' main disadvantage: being a mixed parabolic-hyperbolic system. The new model is achieved by adopting the compressible codeformational time derivative in the stress-strain constitutive relation, resulting in a consistency with the principle of material frame indifference. The main advantage of the new formulation is that the resulting system of equations is purely hyperbolic. The proposed formulation is used to model the benchmark problem of the shock/boundary layer/expansion fan interaction with an apparent degree of success.

We consider the fundamental problem of solving quadratic systems of equations in $n$ variables, where $y_i = |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$, $i = 1, \ldots, m$ and $\boldsymbol{x} \in \mathbb{R}^n$ is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data $\{\boldsymbol{a}_i\}$ and $\{y_i\}$ as soon as the ratio $m/n$ between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have $y_i \approx |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$ and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size., accepted to Communications on Pure and Applied Mathematics (CPAM)

This manuscript presents the formulation and application of the Green’s Discrete Transformation Method (GDTM) for the meshfree simulation of transient diffusion problems, including those with moving boundaries. The GDTM implements a linear combination of time-dependent Green’s basis functions defined on a set of source points to approximate the field in the form of a solution series. A discrete transformation is implemented to evaluate unknown coefficients of this series, which eliminates the need to use time integration schemes. We will study the optimal number and location of the GDTM source points that yield the highest level of accuracy, while maintaining a manageable condition number for the resulting linear system of equations. The optimal values of these parameters, which are inherently independent of the domain geometry, are determined such that the basis functions have appropriate features for approximating the field. A comprehensive convergence study is presented to show the precision and convergence rate of the GDTM for modeling various diffusion problems. We also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies: heat transfer in a turbine blade, thermal response of a porous material, and localized (pitting) corrosion in stainless steel. Copyright c 2015 John Wiley & Sons, Ltd.

Summary This paper presents a coupled hydro-mechanical formulation for the simulation of non-planar three-dimensional hydraulic fractures. Deformation in the rock is modeled using linear elasticity, and the lubrication theory is adopted for the fluid flow in the fracture. The governing equations of the fluid flow and elasticity and the subsequent discretization are fully coupled. A Generalized/eXtended Finite Element Method (G/XFEM) is adopted for the discretization of the coupled system of equations. A Newton–Raphson method is used to solve the resulting system of nonlinear equations. A discretization strategy for the fluid flow problem on non-planar three-dimensional surfaces and a computationally efficient strategy for handling time integration combined with mesh adaptivity are also presented. Several three-dimensional numerical verification examples are solved. The examples illustrate the generality and accuracy of the proposed coupled formulation and discretization strategies. Copyright © 2015 John Wiley & Sons, Ltd.

To reduce the numerical errors arising from the improper enforcement of the artificial boundary conditions on the distant surface that encloses the underground part of the subsurface, we present a finite-element–infinite-element coupled method to significantly reduce the computation time and memory cost in the 2.5D direct-current resistivity inversion. We first present the boundary value problem of the secondary potential. Then, a new type of infinite element is analysed and applied to replace the conventionally used mixed boundary condition on the distant boundary. In the internal domain, a standard finite-element method is used to derive the final system of linear equations. With a novel shape function for infinite elements at the subsurface boundary, the final system matrix is sparse, symmetric, and independent of source electrodes. Through lower upper decomposition, the multi-pole potentials can be swiftly obtained by simple back-substitutions. We embed the newly developed forward solution to the inversion procedure. To compute the sensitivity matrix, we adopt the efficient adjoint equation approach to further reduce the computation cost. Finally, several synthetic examples are tested to show the efficiency of inversion.

The aim of this article is to present and analyze first-order system least-squares spectral method for the Stokes equations in two-dimensional spaces. The Stokes equations are transformed into a first-order system of equations by introducing vorticity as a new variable. The least-squares functional is then defined by summing up the L w 2 - and H w − 1 -norms of the residual equations. The H w − 1 -norm in the least-squares functional is replaced by suitable operator. Continuous and discrete homogeneous least-squares functionals are shown to be equivalent to H w 1 -norm of velocity and L w 2 -norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015

Summary This paper proposes a second-order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L2, H1 error estimates of the velocity and L2 error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ1, different values of null viscosity coefficient μ0 are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ1 are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.

Summary Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B-splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B-splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B-splines should be locally refined. The error estimates are calculated based on recovery of the L2-projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two-dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd.

Summary We propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non-linear damage growth is accurately accounted by a local FEM mesh. An integral-type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non-linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.

SUMMARY We present a fast time-domain multiphysics simulation framework that combines full-wave electromagnetism (EM) and carrier transport in semiconductor devices (technology computer-aided design (TCAD)) for radio frequency (RF) and mixed-signal modules. The proposed framework features a division of linear and nonlinear components in the EM–TCAD coupled system. The linear portion is extracted and handled independently with high efficiency by a matrix exponential approach assisted with Krylov subspace method. The nonlinear component is treated by ordinary Newton's method yet with a much sparser Jacobian matrix that leads to substantial speedup in solving the linear system of equations. More convenient error management and adaptive control are also available through the linear and nonlinear decoupling. Furthermore, a new form of system formulation is developed to further enhance the efficiency of the proposed framework by reducing the stiffness of EM–TCAD systems via special equation and variable transforms. Copyright © 2015 John Wiley & Sons, Ltd.

Summary A model formulated in terms of conserved variables is proposed for its use in the study of internal ballistic problems of pyrotechnical mixtures and propellants. It is a transient two-phase flow model adapted from the non-conservative Gough model. This conversion is mathematically attractive because of the wide range of numerical methods for this kind of systems that may be applied. We propose the use of the AUSM+, AUSM + up and Rusanov schemes as an efficient alternative for this type of two-phase problem. A splitting technique is applied, which solves the system of equations in several steps. A second-order approach based on Monotonic Upstream-Centred Scheme for Conservation Laws (MUSCL) is also used. Some tests are used to validate the code, namely a shock wave test, a contact discontinuity problem and an internal ballistics problem. In this last case, one-dimensional numerical results are compared with experimental data of 155-mm gunshots. Copyright © 2015 John Wiley & Sons, Ltd.

We present a novel multigrid scheme based on a cut-cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved rate of convergence compared to previous methods. Existing fluid solvers with voxelized domains can directly benefit from this approach by only modifying the representation of the non-fluid domain. The necessary building blocks are fully parallelizable and can therefore benefit from multi- and many-core architectures.

An optimized parallel algorithm is proposed to solve the problem occurred in the process of complicated backward substitution of cyclic reduction during solving tridiagonal linear systems. Adopting a hybrid parallel model, this algorithm combines the cyclic reduction method and the partition method. This hybrid algorithm has simple backward substitution on parallel computers comparing with the cyclic reduction method. In this paper, the operation count and execution time are obtained to evaluate and make comparison for these methods. On the basis of results of these measured parameters, the hybrid algorithm using the hybrid approach with a multi-threading implementation achieves better efficiency than the other parallel methods, that is, the cyclic reduction and the partition methods. In particular, the approach involved in this paper has the least scalar operation count and the shortest execution time on a multi-core computer when the size of equations meets some dimension threshold. The hybrid parallel algorithm improves the performance of the cyclic reduction and partition methods by 19.2% and 13.2%, respectively. In addition, by comparing the single-iteration and multi-iteration hybrid parallel algorithms, it is found that increasing iteration steps of the cyclic reduction method does not affect the performance of the hybrid parallel algorithm very much. Copyright © 2015 John Wiley & Sons, Ltd.

A methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function (VRF) in order to compute an a priori low cost estimate of the spatial distribution of the second-order error of the response, as a function of the number of terms used in the truncated Karhunen-Lo eve (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second-order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method (SSFEM). Copyright c © 2014 John Wiley & Sons, Ltd.

Objectives This study explores potential reciprocity in the relationship of democracy with peace. In the decade since introduction of a potential causal role for conflict relative to regime type in the 1990s, a number of theoretical and empirical advances have been made. These steps forward include an innovative statistical technique for estimating multiple equations with categorical dependent variables. Methods We use a system of equations to explore the reciprocal relationship. In our statistical model, we employ a variety of new independent variables and novel measurements of established independent variables that have emerged in the conflict processes literature. Results We find that the reciprocal relationship between conflict and regime type is sustained as a theoretically and empirically valid way to think about these two concepts. Conclusions Our results support the idea that reciprocity should find a place within the continuing trajectory of the neo-Kantian research program.

In this paper, we present a preconditioned variant of the generalized successive overrelaxation (GSOR) iterative method for solving a broad class of complex symmetric linear systems. We study conditions under which the spectral radius of the iteration matrix of the preconditioned GSOR method is smaller than that of the GSOR method and determine the optimal values of iteration parameters. Numerical experiments are given to verify the validity of the presented theoretical results and the effectiveness of the preconditioned GSOR method. Copyright c © 2000 John Wiley & Sons, Ltd.

Mathematics Subject Classification (2010) 74H40, 74M15, 65N30 In this paper we investigate the dynamic behaviour of a thermoelastic diffusion rod clamped at one end and moves freely between two stops at the other. The contact is modelled with the Signorini or normal compliance conditions. The coupled system of equations consists of a hyperbolic equation and two parabolic equations. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. The existence of a weak solution is proved using a penalization method and compensated compactness. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity. We describe the discrete finite element method to our numerical approximations and we show that the given solution converges to the weak solution. Finally, we give an error estimate assuming extra regularity on the solution and we give some results of our numerical experiments.

Summary An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second-order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB1T++AKXBKT=C. The most straightforward approach computes XRmxn from the solution of an mn x mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines greedy low-rank techniques with Galerkin projection and preconditioned gradients. In turn, only linear systems of size m x m and n x n need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as m = n = O(10(5)). Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as we do not need to assume that C has low rank. Copyright (c) 2015 John Wiley & Sons, Ltd.

This paper considers the problem of achieving a very accurate tracking of a pre-specified desired output trajectory , for linear, multiple input multiple output, non-minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desired and the corresponding sought input are partitioned in a transient component ( and ut(k), respectively) and steady-state ( and us(k), respectively). The desired transient component is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of ut(k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least-squares solution of the over-determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady-state input us(k) is directly analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of .

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds-averaged Navier–Stokes and k − ω turbulence model equations for solving all-speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non-preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill-conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean-flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high-lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high-order DG solver at different flow regimes.

Summary We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off-diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.

Spatial statistical analyses are often used to study the link between environmental factors and the incidence of diseases. In modelling spatial data, the existence of spatial correlation between observations must be considered. However, in many situations, the exact form of the spatial correlation is unknown. This paper studies environmental factors that might influence the incidence of malaria in Afghanistan. We assume that spatial correlation may be induced by multiple latent sources. Our method is based on a generalized estimating equation of the marginal mean of disease incidence, as a function of the geographical factors and the spatial correlation. Instead of using one set of generalized estimating equations, we embed a series of generalized estimating equations, each reflecting a particular source of spatial correlation, into a larger system of estimating equations. To estimate the spatial correlation parameters, we set up a supplementary set of estimating equations based on the correlation structures that are induced from the various sources. Simultaneous estimation of the mean and correlation parameters is performed by alternating between the two systems of equations. 2017 Royal Statistical Society We thank the Associate Editor and the referees for their perceptive comments and suggestions, that have led to a greatly improved version of this paper. Denis Leung�s research is funded by the Research Center at Singapore Management University. You-Gan Wang�s research is funded by Australian Research Council discovery grant DP130100766 and project DP160104292. Scopus

Summary This paper studies the performance of different algorithms for solving a dense symmetric indefinite linear system of equations on multicore CPUs with a Graphics Processing Unit (GPU). To ensure the numerical stability of the factorization, pivoting is required. Obtaining high performance of such algorithms on the GPU is difficult because all the existing pivoting strategies lead to frequent synchronizations and irregular data accesses. Until recently, there has not been any implementation of these algorithms on a hybrid CPU/GPU architecture. To improve their performance on the hybrid architecture, we explore different techniques to reduce the expensive data transfer and synchronization between the CPU and GPU, or on the GPU (e.g., factorizing the matrix entirely on the GPU or in a communication-avoiding fashion). We also study the performance of the solver using iterative refinements along with the factorization without pivoting combined with the preprocessing technique based on random butterfly transformations, or with the mixed-precision algorithm where the matrix is factorized in single precision. This randomization algorithm only has a probabilistic proof on the numerical stability, and for this paper, we only focused on the mixed-precision algorithm without pivoting. However, they demonstrate that we can obtain good performance on the GPU by avoiding the pivoting and using the lower precision arithmetics, respectively. As illustrated with the application in acoustics studied in this paper, in many practical cases, the matrices can be factorized without pivoting. Because the componentwise backward error computed in the iterative refinement signals when the algorithm failed to obtain the desired accuracy, the user can use these potentially unstable but efficient algorithms in most of the cases and fall back to a more stable algorithm with pivoting only in the case of the failure. Copyright © 2017 John Wiley & Sons, Ltd.