Laplace-SBA Method for Solving Nonlinear Coupled Burger's Equations

Burger’s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-Somé Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.

Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. This book uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations. The construction itself—an intricate algorithm with hidden symmetries—mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful “Main Lemma”—used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem—has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

On 16th and 32th Order Multioperators-Based Schemes for Smooth and Discontinuous Fluid Dynamics Solutions

The paper presents a novel family of arbitrary high order multioperators approximations for convection, convection-diffusion or the fluid dynamics equations. As particular cases, the 16th- and 32th-order skew-symmetric multioperators for derivatives supplied by the 15th- and 31th-order dissipation multioperators are described. Their spectral properties and the comparative efficiency of the related schemes in the case of smooth solutions are outlined. The ability of the constructed conservative schemes to deal with discontinuous solutions is investigated. Several types of nonlinear hybrid schemes are suggested and tested against benchmark problems.

Methods of Mathematical Modeling Atmospheric Circulation

In this chapter, atmospheric processes are considered on a planetary scale that is impossible to imagine without a detailed three-dimensional modeling based on geophysical fluid dynamics equations of thermodynamics, radiation transfer, the kinetic equations describing the actual movement of air masses, conversion and transfer of energy and matter, the formation of clouds, aerosols, and rainfall and other atmospheric processes.