Excellent Conservation Properties Research Articles

MAGMA: a three-dimensional, Lagrangian magnetohydrodynamics code for merger applications

We present a new, completely Lagrangian magnetohydrodynamics (MHD) code that is based on the smoothed particle hydrodynamics (SPH) method. The equations of self-gravitating hydrodynamics are derived self-consistently from a Lagrangian and account for variable smoothing length (‘grad-h’) terms in both the hydrodynamic and the gravitational acceleration equations. The evolution of the magnetic field is formulated in terms of so-called Euler potentials which are advected with the fluid and thus guarantee the MHD flux-freezing condition. This formulation is equivalent to a vector potential approach and therefore fulfils the ∇·B= 0 constraint by construction. Extensive tests in one, two and three dimensions are presented. The tests demonstrate the excellent conservation properties of the code and show the clear superiority of the Euler potentials over earlier magnetic SPH formulations.

Gradient symplectic algorithms for solving the radial Schrödinger equation

The radial Schrödinger equation for a spherically symmetric potential can be regarded as a one-dimensional classical harmonic oscillator with a time-dependent spring constant. For solving classical dynamics problems, symplectic integrators are well known for their excellent conservation properties. The class of gradient symplectic algorithms is particularly suited for solving harmonic-oscillator dynamics. By use of Suzuki’s rule [Proc. Jpn. Acad., Ser. B: Phys. Biol. Sci. 69, 161 (1993)] for decomposing time-ordered operators, these algorithms can be easily applied to the Schrödinger equation. We demonstrate the power of this class of gradient algorithms by solving the spectrum of highly singular radial potentials using Killingbeck’s method [J. Phys. A 18, 245 (1985)] of backward Newton-Ralphson iterations.

Numerical Simulation of Unsteady Flow in Air-Lift Pump.

A numerical method for simulation of unsteady flow in a large-scale air-lift pump, which is the main element of a deep-sea mining system, has been developed. We have proposed an iterative scheme for implicit time marching of density and pressure, and numerically simulated an air-water two-phase flow using a drift-flux model. In this paper, we describe improvements in the numerical scheme, especially for the convection term. First, convection terms are discretized based on the velocity component of each phase. Second, the finite difference method, which has excellent energy conservation properties, is applied. Then, the upstream-shifted interpolation technique is introduced. Numerical results for both unsteady and steady-state features show reasonable agreement with the experimental data obtained using an air-lift system of a depth of 200 m.

Evolution of long water waves in variable channels

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available.

Numerical Simulation of Unsteady Flow in Air-Lift Pump.

A numerical method for simulation of unsteady flow in an air-lift pump, which is the main element of a deep-sea mining system, has been developed. The authors have proposed an iterative scheme for implicit time marching of density and pressure, and performed a numerical simulation of air-water two-phase flow using a drift -flux model. In the present paper, improvement of the scheme, especially for the convection term, is described. First, convective terms are discretized based on the velocity component of each phase. Second, the finite difference method, which has excellent energy conservation properties, is applied. Then, the upstream-shifted interpolation technique is introduced. Numerical results for both the unsteady features and the steady-state flow field show reasonable agreement with the experimental data obtained with the air-lift system of 200 m depth.