To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances

The article is devoted to numerical methods with strong orders 1.0, 1.5, and 2.0 of convergence for multidimensional dynamical systems with random disturbances. We consider explicit one-step numerical methods for Ito stochastic differential equations. For numerical modeling of iterated Ito stochastic integrals of multiplicities 1 to 4 we using the method of multiple Fourier-Legendre series, converging in the mean in the space $L_2([t, T]^k),$ $k=1,\ldots,4.$ The article is addressed to engineers who use numerical modeling in stochastic control and for solving the nonlinear filtering problem.Comment: 21 pages.Translation into English. Minor change

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. Combined Approach Based on Generalized Multiple and Iterated Fourier Series

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of multiplicity 2 on the base of the combined approach of generalized multiple and iterated Fourier series. We consider two different parts of the expansion of iterated Stratonovich stochastic integrals. The mean-square convergence of the first part is proved on the base of generalized multiple Fourier series converging in the mean-square sense in the space $L_2([t, T]^2).$ The mean-square convergence of the second part is proved on the base of generalized iterated (double) Fourier series converging pointwise. At that, we prove the iterated limit transition for the second part of the expansion on the base of the classical theorems of mathematical analysis. The results of the article can be applied to the numerical integration of Ito stochastic differential equations.Comment: 18 pages. Sect. 3 was added. arXiv admin note: text overlap with arXiv:1801.05654, arXiv:1801.00784, arXiv:1801.01564, arXiv:1712.09746, arXiv:1801.03195, substantial text overlap with arXiv:1712.0951

The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

In this review article we collected more than ten theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author. These theorems open a new direction for study of iterated Ito and Stratonovich stochastic integrals. The expansions based on multiple and iterated Fourier-Legendre series as well as on multiple and iterated trigonomectic Fourier series converging in the mean and pointwise are presented in the article. Some of these theorems are connected with the iterated stochastic integrals of multiplicities 1 to 5. Also we consider two theorems on expansions of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ as well as two theorems on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized iterated Fourier series converging pointwise. On the base of the presented theorems we formulate 3 hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k).$ The mentioned iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. Moreover, the considered expansions from these 3 hypotheses contain only one operation of the limit transition and substantially simpler than their analogues for iterated Ito stochastic integrals. Therefore, the results of the article can be useful for the numerical integration of Ito stochastic differential equations. Also, the results of the article were reformulated in the form of theorems of the Wong-Zakai type for iterated Stratonovich stochastic integrals.Comment: 35 pages. Section 12 was added. arXiv admin note: text overlap with arXiv:1712.09516, arXiv:1712.08991, arXiv:1802.04844, arXiv:1801.00231, arXiv:1712.09746, arXiv:1801.0078

Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series

We consider some versions and generalizations of the approach to expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series. The expansions of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales were obtained. For the iterated stochastic integrals with respect to martingales we have proved two theorems. The first theorem is the generalization of expansion for iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series. The second one is the modification of the first theorem for the case of complete orthonormal with weight $r(t_1)\ldots r(t_k)\ge 0$ systems of functions in the space $L_2([t, T]^k)$ (in the first theorem $r(t_1)\ldots r(t_k)\equiv 1$). Mean-square convergence of the considered expansions is proved. The example of expansion of iterated (double) stochastic integrals with respect to martingales with using the system of Bessel functions is considered.Comment: 37 pages. Minor changes. arXiv admin note: text overlap with arXiv:1712.09746, arXiv:1801.05654, arXiv:1801.01564, arXiv:1712.08991, arXiv:1801.00231, arXiv:1801.03195, arXiv:1712.0951

Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions

The article is devoted to the construction of explicit one-step strong numerical methods with the orders of convergence 2.0 and 2.5 for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the numerical methods based on the unified Taylor-Ito and Taylor-Stratonovich expansions. For the numerical modeling of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 5 we appling the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,\ldots,5$. The article is addressed to engineers who use numerical modeling in stochastic control and for solving the non-linear filtering problem. The article will be interesting to scientists who working in the field of numerical integration of stochastic differential equations.Comment: 31 pages. Minor changes. arXiv admin note: text overlap with arXiv:1801.00231, arXiv:1712.08991, arXiv:1802.00643, arXiv:1801.0886

The Interaction of High-Speed Turbulence with Flames: Global Properties and Internal Flame Structure

We study the dynamics and properties of a turbulent flame, formed in the presence of subsonic, high-speed, homogeneous, isotropic Kolmogorov-type turbulence in an unconfined system. Direct numerical simulations are performed with Athena-RFX, a massively parallel, fully compressible, high-order, dimensionally unsplit, reactive-flow code. A simplified reaction-diffusion model represents a stoichiometric H2-air mixture. The system being modeled represents turbulent combustion with the Damkohler number Da = 0.05 and with the turbulent velocity at the energy injection scale 30 times larger than the laminar flame speed. The simulations show that flame interaction with high-speed turbulence forms a steadily propagating turbulent flame with a flame brush width approximately twice the energy injection scale and a speed four times the laminar flame speed. A method for reconstructing the internal flame structure is described and used to show that the turbulent flame consists of tightly folded flamelets. The reaction zone structure of these is virtually identical to that of the planar laminar flame, while the preheat zone is broadened by approximately a factor of two. Consequently, the system evolution represents turbulent combustion in the thin-reaction zone regime. The turbulent cascade fails to penetrate the internal flame structure, and thus the action of small-scale turbulence is suppressed throughout most of the flame. Finally, our results suggest that for stoichiometric H2-air mixtures, any substantial flame broadening by the action of turbulence cannot be expected in all subsonic regimes.Comment: 30 pages, 9 figures; published in Combustion and Flam

Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations

The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of high-order strong numerical methods for Ito stochastic differential equations. Explicit one-step strong numerical schemes with the orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived. Effective mean-square approximations of iterated Ito and Stratonovich stochastic integrals from these numerical schemes are constructed on the base of the multiple Fourier-Legendre series with multiplicities 1 to 6.Comment: 80 pages. Minor changes. The bibliography has been updated. arXiv admin note: text overlap with arXiv:1807.02190, arXiv:1801.00231, arXiv:1801.01564, arXiv:1712.0899

Expansion of Iterated Stratonovich Stochastic Integrals of Fifth Multiplicity Based on Generalized Multiple Fourier Series

The article is devoted to the construction of expansion of iterated Stratonovich stochastic integrals of fifth multiplicity based on the method of generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The mentioned expansion converges in the mean-square sense and contains only one operation of the limit transition in contrast to its existing analogues. The expansion of iterated Stratonovich stochastic integrals turned out much simpler than the appropriate expansion of iterated Ito stochastic integrals. We use the expansion of the latter as a tool of the proof of the expansion for iterated Stratonovich stochastic integrals. The iterated Stratonovich stochastic integrals are the part of the Taylor-Stratonovich expansion of solutions of Ito stochastic differential equations. That is why the results of the article can be applied to the numerical integrations of Ito stochastic differential equations.Comment: 37 pages. Some additions in Sect. 6. arXiv admin note: substantial text overlap with arXiv:1801.01564, arXiv:1801.05654, arXiv:1712.09516; text overlap with arXiv:1801.03195, arXiv:1801.06501, arXiv:1712.08991, arXiv:1801.00231, arXiv:1712.0974