V. N. Govorukhin

DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

A Meshfree Method for the Analysis of Planar Flows of Inviscid Fluids

A variant of a vortex particle-in-cell method is proposed for computing 2D inviscid incompressible flows in a closed domain. The governing equations are the 2D Euler equations in terms of the stream function and the vorticity, or geophysical models of the atmosphere. The approach is based on an approximation of the vorticity field using its values at a set of fluid particles and the stream function is computed by a Galerkin method. The flow domain is divided into rectangular cells. Vorticity in every cell is interpolated by a third order polynomial. The resultant piecewise continuous polynomial approximation of the vorticity is employed to derive analytically Galerkin’s coefficients of the stream function expansion. A parallel algorithm of the proposed method is also presented. The performance of the method is illustrated by some numerical results dealing with flows trough channels and the dynamics of the multipoles vortex patch calculation.