the steger and warming flux vector splitting implicit scheme is used to numerically solve two dimensional reynolds averaged navier–stokes (rans) equations governing the vortex induced vibration of a flexible riser laterally supported by a spring and a damper. the k–ε model is used as turbulence model to simulate the turbulent flow in the wake of the riser. to update the new position of the rise...

The flux-vector and flux-difference splittings of Steger-Warming, Van Leer and Roe are tested in all possible combinations in the implicit and explicit operators that can be distinguished in implicit relaxation methods for the steady Euler and Navier-Stokes equations. The tests include one-dimensional inviscid nozzle flow, and two-dimensional inviscid and viscous shock reflection. Roe’s splitti...

We prove an error estimate for a Lie–Trotter splitting operator associated with the Schrödinger–Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible Euler–Poisson equation is smooth, the error between the numerical solution and the exact solution is controlled in Sobolev spaces, in a suitable phase/ampli...

In this paper we develop a numerical method to solve Boltzmann like equations of kinetic theory which is able to capture the compressible Navier-Stokes dynamics at small Knudsen numbers. Our approach is based on the micro/macro decomposition technique, which applies to general collision operators. This decomposition is performed in all the phase space and leads to an equivalent formulation of t...

A new result for stability of Markov semigroups is presented. We apply this result to the equation of the passive tracer in a compressible random flow showing that the velocity of a particle converges weakly to some random vector.

A gas-kinetic flux splitting method is developed for the ideal magnetohydrodynamics (MHD) equations. The new scheme is based on the direct splitting of the flux function of the MHD equations with the inclusion of “particle” collisions in the transport process. Consequently, the artificial dissipation in the new scheme is greatly reduced in comparison with the MHD flux vector splitting method. N...

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an auxiliary Cauchy problem for a pseudo-parabolic equation. Unconditionally stable vector additive schemes (splitting schemes) are constructed. Numerical resul...

New exact solutions are obtained for several nonlinear physical equations, namely the Navier–Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schrödinger equation. The solution methods make use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.

A new formulation for the numerical solution of low Mach number compressible flow problems is presented and analyzed. In this formulation the thermal part (energy and species equations) is solved implicitly and decoupled from the momentum equation, whereas the hydrodynamic part (momentumcontinuity) is advanced in time using a high order splitting approach which results in overall high order acc...