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This study aims to present a highly efficient operator-splitting finite element method for the nonlinear two-dimensional/three-dimensional (2D/3D) Allen–Cahn (AC) equation which describes the anti-phase domain coarsening in a binary alloy. This method is presented to overcome the higher storage requirements, computational complexity and the nonlinear term in numerical computation for the 2D/3D AC equation.

In each time interval, the authors first split the original equation into a heat equation and a nonlinear equation. Then, they split the high-dimensional heat equation into a series of one-dimensional (1D) heat equations. By solving each 1D subproblem, the authors obtain a numerical solution for heat equation and take it as an initial for the nonlinear equation, which is solved analytically.

The authors show that the proposed method is unconditionally stable. Finally, various numerical experiments are presented to confirm the high accuracy and efficiency of this method.

A new operator-splitting method is presented for solving the 2D/3D parabolic equation. The 2D/3D parabolic equation is split into a sequence of 1D parabolic equations. In comparison with standard finite element method, the present method can save much central processing unit time. Stability analysis and error estimates are derived and numerical results are presented to support the theoretical analysis.

The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations.

Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement.

The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method.

Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method.

The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3 + 1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries.

The framework of differential‐form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike.

The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric‐incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates.

The present formalism can be used to generate concise formulations of complex field problems. The differential‐form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation.

The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finite‐difference operator‐splitting method, a version of the known SIMPLEC‐like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second‐order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 × 321 grid points, at sequentially halved time steps to ensure grid‐independent results. Three types of flow are shown to exist at Re = 500: a steady‐state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.

A semi‐implicit semi‐Lagrangian mixed finite‐difference finite‐volume model for the shallow water equations on a rotating sphere is considered. The main features of the model are the finite‐volume approach for the continuity equation and the vectorial treatment of the momentum equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi‐implicitly. Discretization of this model led to the introducion, in a previous paper, of a splitting technique which highly reduces the computational effort for the numerical solution. In this paper we solve the full set of equations, without splitting, introducing an ad hoc algorithm. A von Neumann stability analysis of this scheme is performed to establish the unconditional stability of the new proposed method. Finally, we compare the efficiency of the two approaches by numerical experiments on a standard test problem. Results show that, due to the devised algorithm, the solution of the full system of equations is much more accurate while slightly increasing the computational cost.

The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.

The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.

Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.

The approach can be used for the numerical simulation of shallow‐water flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallow‐water models are applied.

The concept of modularity has gained considerable traction in technology studies as a way to conceive, describe, and innovate complex systems, such as product design or organizational structures. In the recent literature, technological modularity has often been intertwined with business model innovation, and scholarship has started investigating how modularity in technology affects changes in business models, both at the cognitive and activity system levels. Yet we still lack a theoretical definition of what modularity is in the business model domain. Business model innovation also encompasses different possibilities of modelling businesses, which are not clearly understood nor classified. We ask when, how, and if modularity theory can be extended to business models in order to enable effective and efficient modelling. We distinguish theoretically between modularity for technology and for business models, and investigate the key processes of modularization and manipulation. We introduce the basic operations of business modelling via modular operators adapted from the technological modularity domain, using iconic examples to develop an analogical reasoning between modularity in technology and in business models. Finally, we discuss opportunities for using modularity theory to foster the understanding of business models and modelling, and develop a challenging research agenda for future investigations.

Some results related to the algorithmic behaviour in semiconductor devices numerical simulations (‐static case‐), using mixed finite elements and operator splitting techniques have been presented in. The drift‐diffusion model written with the electrostatic potential φ and the quasi‐Fermi levels φn and φp is used.

The objective of this work is to empirically investigate the design of a service delivery system that supports the provision of modular service logistics offerings.

An in-depth single-case study relying on interview data and extensive documentary evidence is carried out in the business-to-business (B2B) logistics sector. Three main analytical techniques are used to make sense of the qualitative data: thematic analysis, process mapping and the application of modular operators.

A modular service delivery system comprises three types of processes that collectively deliver modular offerings. The platform consists of core processes that enable the collection, transport and delivery of physical items for all offerings (modular and non-modular). Dedicated modular processes are mandatory and exclusive to individual modular offerings. Optional modular processes are shared across several modular offerings. Interfaces regulate physical (e.g. parcels or parts) and information (e.g. booking data) inputs provided by the customer in order to control the interdependencies within these different process types.

The identification of three process types and their interdependencies provides detailed insights into how managers can design modular logistics services that benefit from economies of scale and meet increasingly variable customer requirements. The importance of well-designed interfaces among the customers, the service offering and the service delivery system is highlighted.

This study extends previous modularity studies in service logistics. It is the first study to apply modular operators to determine the presence of modularity in the service delivery system and to establish the role of different process types in enabling modularity in the service delivery system.