Paul J. Dellar

Incompressible limits of lattice Boltzmann equations using multiple relaxation times

Lattice Boltzmann equations using multiple relaxation times are intended to be more stable than those using a single relaxation time. The additional relaxation times may be adjusted to suppress non-hydrodynamic modes that do not appear directly in the continuum equations, but may contribute to instabilities on the grid scale. If these relaxation times are fixed in lattice units, as in previous work, solutions computed on a given lattice are found to diverge in the incompressible (small Mach number) limit. This non-existence of an incompressible limit is analysed for an inclined one dimensional jet. An incompressible limit does exist if the non-hydrodynamic relaxation times are not fixed, but scaled by the Mach number in the same way as the hydrodynamic relaxation time that determines the viscosity.

Shallow water equations with a complete Coriolis force and topography

This paper derives a set of two-dimensional equations describing a thin inviscid fluid layer flowing over topography in a frame rotating about an arbitrary axis. These equations retain various terms involving the locally horizontal components of the angular velocity vector that are discarded in the usual shallow water equations. The obliquely rotating shallow water equations are derived both by averaging the three-dimensional equations and from an averaged Lagrangian describing columnar motion using Hamilton’s principle. They share the same conservation properties as the usual shallow water equations, for the same energy and modified forms of the momentum and potential vorticity. They may also be expressed in noncanonical Hamiltonian form using the usual shallow water Hamiltonian and Poisson bracket. The conserved potential vorticity takes the standard shallow water form, but with the vertical component of the rotation vector replaced by the component locally normal to the surface midway between the upper...

An interpretation and derivation of the lattice Boltzmann method using Strang splitting

The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank-Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity.

Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere

Starting from Hamiltons principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry that capture the change with latitude of the angle between the rotation vector and the local vertical. Being derived using Hamiltons principle, the different β-plane approximations each conserve energy, angular momentum and potential vorticity. They differ in their treatments of the locally horizontal component of the rotation vector, the component that is usually neglected under the traditional approximation. In particular, we derive an extended set of β-plane equations in which the locally vertical and locally horizontal components of the rotation vector both vary linearly with latitude. This was previously thought to violate conservation of angular momentum and potential vorticity. We show that the difficulty in maintaining these conservation laws arises from the need to express the rotation vector as the curl of a vector potential while approximating the true spherical metric by a flat Cartesian metric. Finally, we derive depth-averaged equations on our extended β-plane with topography, and show that they coincide with the extended non-traditional shallow-water equations previously derived in Cartesian geometry.

Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields

The Hamiltonian structure of the inhomogeneous layer models for geophysical fluid dynamics devised by Ripa [Geophys. Astrophys. Fluid Dyn. 70, 85 (1993)] involves the same Poisson bracket as a Hamiltonian formulation of shallow water magnetohydrodynamics in velocity, height, and magnetic flux function variables. This Poisson bracket becomes the Lie–Poisson bracket for a semidirect product Lie algebra under a change of variables, giving a simple and direct proof of the Jacobi identity in place of Ripa’s long outline proof. The same bracket has appeared before in compressible and relativistic magnetohydrodynamics. The Hamiltonian is the integral of the three dimensional energy density for both the inhomogeneous layer and magnetohydrodynamic systems, which provides a compact derivation of Ripa’s models.

Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics

Shallow water magnetohydrodynamics is a recently proposed model for a thin layer of incompressible, electrically conducting fluid. The velocity and magnetic field are taken to be nearly two dimensional, with approximate magnetohydrostatic balance in the perpendicular direction. In this paper a Hamiltonian description, with the ubiquitous noncanonical Lie–Poisson bracket for barotropic magnetohydrodynamics, is derived by integrating the three-dimensional energy density in the perpendicular direction. Specialization to two dimensions yields an elegant form of the bracket, from which further conserved quantities (Casimirs) of shallow water magnetohydrodynamics are derived. These Casimirs closely resemble the Casimirs of incompressible reduced magnetohydrodynamics, so the stability properties of the two systems may be expected to be similar. The shallow water magnetohydrodynamics system is also cast into symmetric hyperbolic form. The symmetric and Hamiltonian properties become incompatible when the relevant ...

Dispersive shallow water magnetohydrodynamics

Shallow water magnetohydrodynamics (SWMHD) is a recently proposed model for a thin layer of incompressible, electrically conducting fluid. The velocity and magnetic field are taken to be nearly two dimensional, with approximate magnetohydrostatic balance in the perpendicular direction, leading to a reduced two-dimensional model. The SWMHD equations have been found previously to admit unphysical cusp-like singularities in finite amplitude magnetogravity waves. This paper extends the Hamiltonian formulation of SWMHD to construct a dispersively regularized system, analogous to the Green–Naghdi equations of hydrodynamics, that supports smooth solitary waves and cnoidal wave trains, and shares the potential vorticity conservation properties of SWMHD.

Lattice Boltzmann algorithms without cubic defects in Galilean invariance on standard lattices

The vast majority of lattice Boltzmann algorithms produce a non-Galilean invariant viscous stress. This defect arises from the absence of a term in the third moment, the equilibrium heat flow tensor, proportional to the cube of the fluid velocity. This moment cannot be specified independently of the lower moments on the standard lattices such as D2Q9, D3Q15, D3Q19 or D3Q27. A partial correction has recently been demonstrated that restores some of these missing cubic terms on the D2Q9 and D3Q27 tensor product lattices. This correction restores Galilean invariance for shear flows aligned with the coordinate axes, but flows inclined at arbitrary angles may show larger errors than before. These remaining errors are due to the diagonal terms of the equilibrium heat flow tensor, which cannot be corrected on standard lattices. However, the remaining errors may be largely absorbed by introducing a matrix collision operator with velocity-dependent collision rates for the diagonal components of the momentum flux tensor. This completely restores Galilean invariance for flows with uniform density, and in general reduces the magnitude of the defect in Galilean invariance from Mach number cubed to Mach number to the fifth power. The effectiveness of the resulting algorithm is demonstrated by comparisons with the standard and partially corrected lattice Boltzmann algorithms for two- and three-dimensional flows.

Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier–Maxwell slip boundary conditions

We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself.

Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.