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Abstract In the present work, we first introduce a general framework for modelling complex multiscale fluids and then focus on the derivation and analysis of a new hybrid continuum-kinetic model. In particular, we combine conservation of mass and momentum for an isentropic macroscopic model with a kinetic representation of the microscopic behavior. After introducing a small scale of interest, we compute the complex stress tensor by means of the Irving-Kirkwood formula. The latter requires an expansion of the kinetic distribution around an equilibrium state and a successive homogenization over the fast in time and small in space scale dynamics. For a new hybrid continuum-kinetic model the results of linear stability analysis indicate a conditional stability in the relevant low speed regimes and linear instability for high speed regimes for higher modes. Extensive numerical experiments confirm that the proposed multiscale model can reflect new phenomena of complex fluids not being present in standard Newtonian fluids. Consequently, the proposed general technique can be successfully used to derive new interesting systems combining the macro and micro structure of a given physical problem. 

In this paper, we develop new high-order numerical methods for hyperbolic systems of non- linear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essen- tially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving. 

We develop a new second-order unstaggered semidiscrete path-conservative central- upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, it does not rely on any (approximate) Riemann problem solver, and it robustly produces high- resolution and nonoscillatory results. The derivation of the scheme is based on the Godunov-Powell nonconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the modified systems with the evolution equations for the corresponding derivatives of the magnetic field components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic field, which guarantees a nonoscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. We test the proposed PCCU scheme on several benchmarks for both ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution, but also to preserve the positivity of computed quantities such as density, pressure, and water depth. 

In this paper, we present a novel approach for simulating solutions of partial differential equations using neural networks. We consider a time-stepping method similar to the finite-volume method, where the flux terms are computed using neural networks. To train the neural network, we collect 'sensor' data on small subsets of the computational domain. Thus, our neural network learns the local behavior of the solution rather than the global one. This leads to a much more versatile method that can simulate the solution to equations whose initial conditions are not in the same form as the initial conditions we train with. Also, using sensor data from a small portion of the domain is much more realistic than methods where a neural network is trained using data over a large domain. 

We study numerical approximations of the reactive Euler equations of gas dynamics. In addition to shock, contact and rarefaction waves, these equations admit detonation waves appearing at the interface between different fractions of the reacting species. It is well-known that in order to resolve the reaction zone numerically, one has to take both space and time stepsizes to be proportional to the reaction time, which may cause the numerical method to become very computationally expensive or even impractical when the reaction is fast. Therefore, it is necessary to develop underresolved numerical methods, which are capable of accurately predicting locations of the detonation waves without resolving their detailed structure. One can distinguish between two different degrees of stiffness. In the stiff case, the reaction time is very small, while in the extremely stiff case, the reaction is assumed to occur instantaneously. In [A. Kurganov, in Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003], we proposed a simple underresolved method—an accurate deterministic projection (ADP) method—for one-dimensional hyperbolic systems with stiff source terms including the reactive Euler equations in the extremely stiff regime. In this paper, we extend the ADP method to the (non-extremely) stiff case, multispecies detonation models, and the two-dimensional reactive Euler equations in all of the aforementioned regimes. We also investigate ways to distinguish between different regimes in practice as well as study the limitations of the proposed ADP methods with respect to the ignition temperature. We demonstrate the accuracy and robustness of the ADP methods in a number of numerical experiments with both relatively low and large ignition temperature, and illustrate the difficulties one may face when the ignition temperature is low. 

In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force, which is proportional to the relative surplus of the cell density compared to the water density. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model for simulating bioconvection in complex geometries. The drop domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. The original chemotaxis-fluid system is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the physical interface. We show that the cf-DD model converges to the chemotaxis-fluid model asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several interesting chemotactic phenomena in sessile drops of different shapes, where the bacterial patterns depend on the droplet geometries.