Scalar Problem Research Articles

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Reducing the complexity of equilibrium problems and applications to best approximation problems

"We consider the scalar equilibrium problems governed by a bifunction in a finite-dimensional framework and we characterize the solutions by means of extreme or exposed points."

Efficient hyperbolic–parabolic models on multi‐dimensional unbounded domains using an extended DG approach

AbstractWe introduce an extended discontinuous Galerkin discretization of hyperbolic–parabolic problems on multidimensional semi‐infinite domains. Building on previous work on the one‐dimensional case, we split the strip‐shaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and nonlinear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semi‐infinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard single‐domain discontinuous finite element techniques.

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

Rymano–Hilberto kraštinis uždavinys daugiamatėms elipsinėms sistemoms

Multidimensional analogues of Cauchy–Riemann quations and of Riemann–Hilbert boundary value problem are studied. Their relation to the scalar boundary value problem is demonstrated.

Optimality conditions and DC-Dinkelbach-type algorithm for vector fractional programming problems with ratios of difference of convex functions

In this work, necessary optimality conditions of KKT type for (weak) Pareto optimality are derived and a DC-Dinkelbach-type algorithm is proposed for vector fractional mathematical programs with ratios of difference of convex (DC) functions, and DC constraints, by reducing the latter to a system of scalar parametric problems and using DC tools. The special case of vector fractional programs with ratios of convex functions is also analysed.

An efficient HLL-based scheme for capturing contact-discontinuity in scalar transport by shallow water flow

An efficient numerical scheme is developed for coupled hydrodynamic and scalar transport systems to guarantee the conservation and the positivity-preserving properties for water depth and scalar concentration. A second-order well-balanced positivity-preserving central-upwind scheme based on the finite volume method is adopted to discretize both the Saint-Venant system and the advective fluxes in the scalar transport system. In particular, an anti-diffusion modification is augmented in an ad hoc manner to the Harten–Lax–van Leer approximate Riemann solver for the scalar transport system, with the aim of significantly reducing the numerical diffusion near contact discontinuities. The proposed scheme is validated through seven numerical experiments, wherein the advection and diffusion processes of scalar transport are considered either separately or in combination. Accuracy is measured based on the difference between the numerical results and analytical solutions, and the performance of the proposed scheme is assessed by comparing it with that of the existing scheme. Convergence analysis confirms that the proposed scheme is accurate to the second order. The stationary steady state with wet-dry fronts is simulated, which verifies that the proposed scheme can preserve the exact C-property. The results for the pure diffusion case reveal that the original scheme has limitations in precisely predicting scalar diffusion when the diffusion coefficient is small. By contrast, the proposed scheme accurately approximates scalar diffusion, even at low diffusivity. The results under flow conditions with various Froude and Péclet numbers confirm that the proposed scheme is more accurate than the existing scheme in solving scalar transport problems over a wide range of shallow water flows.

A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems

A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGNR or GMRES required smaller numbers of iterations with CHDG.

Potentiality of the HOFiD_bvp code in solving different kind of second order boundary value problems

The aim of this paper is to present the potentiality of the code HOFiD_bvp based on high order finite difference, following the approach of the boundary value methods and the upwind methods. Strategies, as variable stepsize and variable order, that helps to handle with solutions of singular perturbation problems, so as the continuation strategy applied for solving nonlinear problems, will be discussed. Now, the method is suitable to solve second order scalar boundary value problems, not only singularly perturbed with one and two parameters, but also problems with discontinuous source terms. Concerning that point some numerical tests will be taken in consideration in order to show the potentiality of the code.

Necessary optimality conditions for a semivectorial bilevel optimization problem via a revisited Charnes–Cooper scalarization

In this paper, we investigate a multiobjective bilevel optimization problem with a vector-valued lower-level objective function. We revisit the Charnes–Cooper scalarization technique for multi-objective programs and use optimal value reformulation to transform the scalar problem into a one-level optimization problem. Using Mordukhovich's generalized differentiation calculus, a new subdifferential estimate and Lipschitz condition for the optimal value function of this problem are developed as a result of this reformulation. In addition, we construct first-order necessary optimality conditions in the smooth setting.

A trust-region approach for computing Pareto fronts in multiobjective optimization

Multiobjective optimization is a challenging scientific area, where the conflicting nature of the different objectives to be optimized changes the concept of problem solution, which is no longer a single point but a set of points, namely the Pareto front. In a posteriori preferences approach, when the decision maker is unable to rank objectives before the optimization, it is important to develop algorithms that generate approximations to the complete Pareto front of a multiobjective optimization problem, making clear the trade-offs between the different objectives. In this work, an algorithm based on a trust-region approach is proposed to approximate the set of Pareto critical points of a multiobjective optimization problem. Derivatives are assumed to be known, allowing the computation of Taylor models for the different objective function components, which will be minimized in two main steps: the extreme point step and the scalarization step. The goal of the extreme point step is to expand the approximation to the Pareto front, by moving towards the extreme points of it, corresponding to the individual minimization of each objective function component. The scalarization step attempts to reduce the gaps on the Pareto front, by solving adequate scalarization problems. The convergence of the method is analyzed and numerical experiments are reported, indicating the relevance of each feature included in the algorithmic structure and its competitiveness, by comparison against a state-of-art multiobjective optimization algorithm.

From black box to clear box: A hypothesis testing framework for scalar regression problems using deep artificial neural networks

Despite the impressive predictive performance exhibited by deep learning across various domains, its application in research models within the social and behavioral sciences has been limited. Deep learning lacks human-accessible interpretability and does not provide statistical inferences. To address these limitations, this article presents a novel model-agnostic hypothesis testing framework tailored to scalar regression problems using deep artificial neural networks. The new framework not only determines each input variable’s direction of influence and statistical significance, but computes effect size measures akin to Cohen’s f2 in traditional ordinary least squares (OLS) regression models. Effect sizes are an important complement to null hypothesis significance testing by providing a practical significance measure that is independent of sample size considerations. To showcase the usefulness of the new framework, its application is demonstrated on both an artificial data set and a social survey using a Python sandbox implementation.

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Unboundedness phenomenon in a model of urban crime

We show that spatial patterns (“hotspots”) may form in the crime model [Formula: see text] which we consider in [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text], [Formula: see text] and initial data [Formula: see text], [Formula: see text] with sufficiently large initial mass [Formula: see text]. More precisely, for each [Formula: see text] and fixed [Formula: see text], [Formula: see text] and (large) [Formula: see text], we construct initial data [Formula: see text] exhibiting the following unboundedness phenomenon: Given any [Formula: see text], we can find [Formula: see text] such that the first component of the associated maximal solution becomes larger than [Formula: see text] at some point in [Formula: see text] before the time [Formula: see text]. Since the [Formula: see text] norm of [Formula: see text] is decreasing, this implies that some heterogeneous structure must form. We do this by first constructing classical solutions to the nonlocal scalar problem [Formula: see text] from the solutions to the crime model by taking the limit [Formula: see text] under the assumption that the unboundedness phenomenon explicitly does not occur on some interval [Formula: see text]. We then construct initial data for this scalar problem leading to blow-up before time [Formula: see text]. As solutions to the scalar problem are unique, this proves our central result by contradiction.

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws.

The Half-Space Matching method for elastodynamic scattering problems in unbounded domains

In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters.

Dynamics of a diffusive mussel-algae system in closed advective environments

In this paper, we are concerned with a diffusive system modeling the interaction of mussel and algae in the water layer overlying the mussel bed, where the algae are the main food source for mussels, and the advection pushes algaes in one direction but not out of the domain. We present a threshold result on the global extinction and persistence of mussels. It shows that the condition for persistence depends on the principal eigenvalue of a scalar boundary value problem, which is related to the diffusion, the speed of the tidal flow, the conversion rate of algae to mussel production, and mussel mortality rate. We further investigate the asymptotic profile of the positive steady state when it exists.

Analysis and evaluation of steady-state and non-steady-state preserving operator splitting schemes for reaction-diffusion(-advection) problems

Reaction-diffusion(-advection) problems are well-known in chemical engineering and computational fluid dynamics. A common feature of these systems is a linear (or non-stiff) transport sub-system and a non-linear (or stiff), highly coupled chemistry sub-system. The expected numerical effort often prohibits a fully coupled solution of the system. Therefore, the system is split into a transport and a reaction sub-system, and each sub-system is solved using specialized solvers. Operator splitting schemes are required to reconstruct the solution of the initial system from the sub-system solutions. Steady-state preserving splitting schemes are particularly essential for steady-state calculations since local time stepping (LTS) or other methods based on fictional time rely on large time steps to be efficient. This work formally analyzes common splitting schemes for reaction-diffusion problems by stability analysis and checking the steady-state preservation of a representative linear scalar problem. Balanced and Simpler Balanced splitting are the only steady-state conservative schemes analyzed. Three new steady-state preserving splitting schemes are proposed based on the findings of the formal analysis. To achieve steady-state preservation, the new schemes use splitting constants based on either the mixing derivative or the chemistry derivative. The formal analysis is accompanied by a dimensionless perfectly stirred reactor (PSR) case and a hydrogen combustion case, both known to be challenging for operator splitting schemes. The test case results are in line with the theoretical results but indicate that the scalar linear analysis is nonviable to capture the full effects of the chemical sub-system. The Simpler and the newly proposed Consistent Staggered splitting schemes give significantly better results than the remaining ones while being second-order and first-order accurate, respectively. If temporal accuracy is irrelevant, e.g., for steady-state solvers, the proposed Consistent Adaptive splitting scheme is promising since it preserves the steady-state solution first-order time accurate with less function evaluations.

An Optimal Control-Based Numerical Method for Scalar Transmission Problems with Sign-Changing Coefficients

An Optimal Control-Based Numerical Method for Scalar Transmission Problems with Sign-Changing Coefficients