Simulation of combustion of methane-air mixture in two-dimensional approximation

The paper presents a mathematical model and the results of a numerical study of the flame propagation of a methane-air mixture. The physical and mathematical formulation of the problem takes into account the thermal expansion of the gas and its subsequent movement. The problem was solved numerically using the Van Leer method to determine the fluxes at the boundaries of the computational cells. A study of flame propagation in a methane-air mixture with a methane content less than or equal to stoichiometric has been carried out. The conditions for focal ignition of a reactive mixture are determined. The influence of the channel walls on the features of flame propagation is shown. The necessity of taking into account the non-isobaricity of the combustion process at the initial stage is demonstrated.

Magnetoconvection in a horizontal duct flow at very high Hartmann and Grashof numbers

Direct numerical simulations and linear stability analysis are carried out to study mixed convection in a horizontal duct with constant-rate heating applied at the bottom and an imposed transverse horizontal magnetic field. A two-dimensional approximation corresponding to the asymptotic limit of a very strong magnetic field effect is validated and applied, together with full three-dimensional analysis, to investigate the flow's behaviour in the previously unexplored range of control parameters corresponding to typical conditions of a liquid metal blanket of a nuclear fusion reactor (Hartmann numbers up to $10^4$ and Grashof numbers up to $10^{10}$ ). It is found that the instability to quasi-two-dimensional rolls parallel to the magnetic field discovered at smaller Hartmann and Grashof numbers in earlier studies also occurs in this parameter range. Transport of the rolls by the mean flow leads to magnetoconvective temperature fluctuations of exceptionally high amplitudes. It is also demonstrated that quasi-two-dimensional structure of flows at very high Hartmann numbers does not guarantee accuracy of the classical two-dimensional approximation. The accuracy deteriorates at the highest Grashof numbers considered in the study.

Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by $$O(d\,{\varepsilon }^{-(4+\delta )})$$ O ( d ε - ( 4 + δ ) ) for any $$\delta \in (0,\infty )$$ δ ∈ ( 0 , ∞ ) under suitable assumptions, where $$d\in {{\mathbb {N}}}$$ d ∈ N is the dimensionality of the problem and $${\varepsilon }\in (0,\infty )$$ ε ∈ ( 0 , ∞ ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

Degenerated and Competing Dirichlet Problems with Weights and Convection

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated (p,q)-Laplacian with weights and a competing (p,q)-Laplacian with weights, respectively. The notion of competing (p,q)-Laplacians with weights is considered for the first time. We present existence and approximation results that hold under the same set of hypotheses on the convection term for both problems. The proofs are based on weighted Sobolev spaces, Nemytskij operators, a fixed point argument and finite dimensional approximation. A detailed example illustrates the effective applicability of our results.

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from the data, a method which can be seen as a generalization of the Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.

COMPUTATION COMPLEXITY OF DEEP RELU NEURAL NETWORKS IN HIGH-DIMENSIONAL APPROXIMATION

The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\"older-Nikol'skii spaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.

DISSIPATIVE INSTABILITIES AND SUPERRADIATION REGIMES (CLASSIC MODELS)

The generation of an electromagnetic field by oscillators in an open resonator is discussed in a one-dimensional approximation. In this case, the development of the so-called dissipative instability  the dissipative generation regime. Such an instability with the generation of electromagnetic oscillations arises when the decrement of oscillations in an open resonator in the absence of oscillators turns out to be greater than the increment of the resulting instability of the system of oscillators placed in this resonator. It is assumed that the oscillators do not interact with each other, and only the resonator field affects their behavior. If the resonator field is absent or small, the superradiance regime is possible, when the radiation of each oscillator is essential and the field in the system is the sum of all the eigenfields of the oscillators. In the dissipative regime of instability generation, the system of oscillators is synchronized by the induced resonator field. The synchronization of the oscillators in the superradiance mode owes its existence to the integral field of the entire system of oscillators. With a weak nonlinearity of the oscillators, a small initiating external field is required to excite the generation regime. It is noteworthy that the maximum value of the superradiance field is approximately two times less than the maximum field that the same particles could generate if they were at the same point. In all cases, for a given open resonator, the superradiance field turned out to be somewhat larger than the resonator field. Nevertheless, for the same resonator, the increments and attainable field amplitudes in both cases are of the same order of magnitude.