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The unsteady incompressible Navier–Stokes equations in primitive variables are often numerically solved by segregating the computation of velocity and pressure, according to either functional analysis arguments following the pioneering work of A.J. Chorin and R. Temam (STD, Split-Then-Discretize paradigm) or linear algebra arguments based on the inexact block factorization of the discretized problem (DTS, Discretize-Then-Split paradigm). The presence of the time derivative allows for the calibration of an appropriate approximation of the pseudo-differential operator of the pressure problem and excellent results in terms of both accuracy and efficiency have been obtained as witnessed by the abundant literature. The extension of the same segregated approach to the steady Navier–Stokes equations is unclear, unless a pseudo-time advancing formulation is undertaken. In this paper we present a methodology for a segregated computation of the primitive variables in a genuinely steady formulation, so to avoid iterations to get to the steady limit. The approach is largely inspired by the algebraic factorization of the unsteady problem (DTS approach), yet we detail specific settings required by the absence of the velocity time-derivative. The basic idea relies on the introduction of some parameters in a modified Picard linearization. We discuss stability bounds and the convergence of the segregated method to the unsplit solution. Several numerical results on different test cases confirm the efficiency of the procedure. [ABSTRACT FROM AUTHOR]

The outbreak of COVID-19, beginning in 2019 and continuing through the time of writing, has led to renewed interest in the mathematical modeling of infectious disease. Recent works have focused on partial differential equation (PDE) models, particularly reaction–diffusion models, able to describe the progression of an epidemic in both space and time. These studies have shown generally promising results in describing and predicting COVID-19 progression. However, people often travel long distances in short periods of time, leading to nonlocal transmission of the disease. Such contagion dynamics are not well-represented by diffusion alone. In contrast, ordinary differential equation (ODE) models may easily account for this behavior by considering disparate regions as nodes in a network, with the edges defining nonlocal transmission. In this work, we attempt to combine these modeling paradigms via the introduction of a network structure within a reaction–diffusion PDE system. This is achieved through the definition of a population-transfer operator, which couples disjoint and potentially distant geographic regions, facilitating nonlocal population movement between them. We provide analytical results demonstrating that this operator does not disrupt the physical consistency or mathematical well-posedness of the system, and verify these results through numerical experiments. We then use this technique to simulate the COVID-19 epidemic in the Brazilian region of Rio de Janeiro, showcasing its ability to capture important nonlocal behaviors, while maintaining the advantages of a reaction–diffusion model for describing local dynamics. [ABSTRACT FROM AUTHOR]

Dynamic Mode Decomposition (DMD) is an unsupervised machine learning method that has attracted considerable attention in recent years owing to its equation-free structure, ability to easily identify coherent spatio-temporal structures in data, and effectiveness in providing reasonably accurate predictions for certain problems, particularly over short-to-medium time frames. Despite these successes, the application of DMD to certain problems featuring highly nonlinear transient dynamics remains challenging. In such cases, DMD may not only fail to provide acceptable predictions but may indeed fail to recreate the data in which it was trained, restricting its application to diagnostic purposes (i.e., feature identification and extraction). For many such problems in the biological and physical sciences, the structure of the system obeys a compartmental framework, in which the transfer of mass, energy, or some other quantity of interest within the system moves across states. In these cases, the behavior of the system may not be accurately recreated by applying DMD to a single quantity within the system, as proper knowledge of the system dynamics, even for a single compartment, requires that the behavior of other compartments is taken into account in the DMD process. In the present work, we demonstrate, theoretically and numerically, that, when performing DMD on a fully coupled PDE system with compartmental structure, one may recover useful predictive behavior, even when DMD performs poorly when acting compartment-wise. We also establish that important physical quantities, such as mass conservation, are maintained in the coupled-DMD extrapolation. The mathematical and numerical analysis suggests that DMD, properly applied, may be a powerful tool for this common class of problems In particular, we show interesting numerical applications to a continuous delayed-SIRD model for Covid-19, and to a problem from additive manufacturing considering a nonlinear temperature field and the resulting change of material phase from powder, liquid, and solid states. [ABSTRACT FROM AUTHOR]

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a conforming, highly non-uniform mesh numerically unfeasible. In fact, such meshes may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. The problem becomes even more prohibitive when the region requiring a finer-level mesh changes in time, requiring the introduction of refinement and derefinement techniques. To address the aforementioned challenges, we employ a technique related to the Fat boundary method as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings with a series of two-dimensional numerical tests. • New method for solving problems with localized nonhomogeneous material properties. • Splits problem into two levels, a coarse global mesh and a fine local mesh. • Allows for use of fine mesh without nonuniform meshes or refinement/derefinement. • Several mathematical results establish consistency of the two-level formulation. • Effectiveness demonstrated on a series of numerical tests. [ABSTRACT FROM AUTHOR]