V. M. Goloviznin

Compact Accurately Boundary-Adjusting high-REsolution Technique for fluid dynamics

A novel high-resolution numerical method is presented for one-dimensional hyperbolic problems based on the extension of the original Upwind Leapfrog scheme to quasi-linear conservation laws. The method is second-order accurate on non-uniform grids in space and time, has a very small dispersion error and computational stencil defined within one space-time cell. For shock-capturing, the scheme is equipped with a conservative non-linear correction procedure which is directly based on the maximum principle. Plentiful numerical examples are provided for linear advection, quasi-linear scalar hyperbolic conservation laws and gas dynamics and comparisons with other computational methods in the literature are discussed.

The principle of minimum of partial local variations for determining convective flows in the numerical solution of one-dimensional nonlinear scalar hyperbolic equations

For the CABARET finite difference scheme, a new approach to the construction of convective flows for the one-dimensional nonlinear transport equation is proposed based on the minimum principle of partial local variations. The new approach ensures the monotonicity of solutions for a wide class of problems of a fairly general form including those involving discontinuous and nonconvex functions. Numerical results illustrating the properties of the proposed method are discussed.

CABARET scheme in velocity-pressure formulation for two-dimensional incompressible fluids

The CABARET method was generalized to two-dimensional incompressible fluids in terms of velocity and pressure. The resulting algorithm was verified by computing the transport and interaction of various vortex structures: a stationary and a moving solitary vortex, Taylor-Green vortices, and vortices formed by the instability of double shear layers. Much attention was also given to the modeling of homogeneous isotropic turbulence and to the analysis of its spectral properties. It was shown that, regardless of the mesh size, the slope of the energy spectra up to the highest-frequency harmonics is equal −3, which agrees with Batchelor’s enstrophy cascade theory.

Cabaret scheme for two-dimensional incompressible fluid in terms of the stream function-vorticity variables

In the present work, the Cabaret method is generalized to the case of two-dimensional incompressible fluid in terms of the stream function-vorticity variables. Using the example of the solitary vortex problem, the high quality of the obtained algorithm in terms of its dissipative and dispersion properties is demonstrated. In the problem about decaying homogeneous isotropic turbulence, the slopes of the energy spectra for all grids (16 × 16, 32 × 32, 64 × 64, 128 × 128) are (−3) up to the highest harmonics, which coincides with Batchelor’s theory.

Generalization of the CABARET scheme to two-dimensional orthogonal computational grids

It is proposed to generalize the CABARET method to a two-dimensional system of Euler equations. The transition from one-dimensional problems of gas dynamics to multidimensional ones for the CABARET scheme involves a number of innovative aspects. The first is a procedure for spatial splitting of an algorithm in order to calculate new flow variables. The second one is the specific application of the maximum principle aimed at regularizing the solutions for inhomogeneous equations of transfer of local invariants in different directions. Examples are given of test and model calculations.

New two-level leapfrog scheme for modeling the stochastic Landau-Lifshitz equations

A two-level modification of the classical nondissipative leapfrog scheme with nonlinear flux correction has been developed for fluctuating hydrodynamics problems. The new algorithm has shown to satisfy the fluctuation-dissipation theorem to high accuracy. The results of various numerical tests, including equilibrium, nonequilibrium, one-, and two-dimensional systems, are compared with theoretical predictions, direct molecular simulations, and results produced by MacCormack’s schemes, the piecewise parabolic method, and a third-order Runge-Kutta scheme. The new algorithm is well suited for parallel computations due to its implementation simplicity and compact stencil.

Direct modeling of the interaction between vortex pairs

This work is devoted to the numerical modeling of the conjugate problem on the dynamics of two vortex pairs and calculation of the acoustic field generation at Reynolds numbers Re = 5000–10 000 on a fixed Eulerian grid. The computation was based on the CABARET scheme. For the main characteristics of integrated solutions, such as the slip period of vortex pairs and the velocity of their center of mass, a comparison is presented with the analytical solution obtained for the case of point vortices in an ideal fluid. The sensitivity of the obtained numerical solutions to the grid refinement was studied including both the hydrodynamic near field and instantaneous and root-mean-square acoustic pulsations.

CABARET scheme for the numerical solution of aeroacoustics problems: Generalization to linearized one-dimensional Euler equations

A generalization of the CABARET finite difference scheme is proposed for linearized one-dimensional Euler equations based on the characteristic decomposition into local Riemann invariants. The new method is compared with several central finite difference schemes that are widely used in computational aeroacoustics. Numerical results for the propagation of an acoustic wave in a homogeneous field and the refraction of this wave through a contact discontinuity obtained on a strongly nonuniform grid are presented.

CABARET scheme in vorticity-velocity variables for the numerical modeling of ideal fluid motion in a two-dimensional domain

A modification of the CABARET scheme is proposed for the numerical solution of equations of ideal fluid motion in vorticity-velocity variables. The dissipative and dispersive properties of the obtained numerical algorithm were investigated for the problem of an isolated vortex. Calculations were performed for decaying homogeneous isotropic turbulence on grids of varying density. In all investigated grids, the spectral density of the kinetic energy was found to obey the “-3” law, which conforms to the Kraichnan-Batchelor theory. The structural functions of the obtained vortex flow conform to the law derived using the dimension theory.

High-resolution numerical algorithm for one-dimensional scalar conservation laws with a constrained solution

The CABARET computational algorithm is generalized to one-dimensional scalar quasilinear hyperbolic partial differential equations with allowance for inequality constraints on the solution. This generalization can be used to analyze seepage of liquid radioactive wastes through the unsaturated zone.