The goal of this research is developing a unified numerical method for simulating continuum and transitional flow. To achieve our ultimate goal, first, hyperbolic-relaxation equations are introduced, then a new discretization method is developed. The method is based on Huynh’s upwind moment scheme, with implicit treatment of the source term. Our previous linear method is generalized to 1-D nonl...

As a numerical method for solving ordinary differential equations y′ = f(x, y), the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for f(x, y) to a finite system that is sufficient and necessary for the improved ...

Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables (θ, φ, ψ) these integrals give separation equa...

In this paper, an algorithm is proposed to avoid singularity associated with the most famous minimum element attitude parametrization, Euler angle set. The proposed algorithm makes use of the method of sequential rotation to avoid singularity associated with Euler angle set. Further, a switching algorithm is also proposed to switch between different Euler angle sets to avoid the singularity whi...

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with...

The general class of problems we consider is the following: Let Ω1 be a bounded domain in R d for d ≥ 2 and let u be a velocity field on all of R. Suppose that for all R ≥ 1 we have an operator TR that projects u restricted to RΩ1 (Ω1 scaled by R) into a function space on RΩ1 for which the solution to some initial value problem is well-posed with TRu 0 as the initial velocity. Can we show that ...

We give a construction of a divergence-free vector field u0 ∈ H s ∩ B ∞,∞ , for all s < 1/2, such that any Leray-Hopf solution to the Navier-Stokes equation starting from u0 is discontinuous at t = 0 in the metric of B ∞,∞ . For the Euler equation a similar result is proved in all Besov spaces B r,∞ where s > 0 if r > 2, and s > n(2/r − 1) if 1 ≤ r ≤ 2.

We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.

We investigate discrete kinetic models in the Fluid dynamic limit described by the Euler system and the Navier-Stokes correction obtained by the Chapman Enskog procedure. We show why reliable “small” systems can be expected only for small Mach numbers and derive a calculus for designing models for given Prandtl numbers.