The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is very challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible ...

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term inv...

We develop the symbolic representation method to derive the hierarchies of (2+1)-dimensional integrable equations from the scalar Lax operators and to study their properties globally. The method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in C or a noncommutative associative algebra. We prove that these hierarchies are indeed qu...

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the beha...

It is well known that the Boussinesq equation is the bidirectional equivalent of the celebrated Korteweg-de Vries equation. Here we consider Boussinesq-type versions of two classical unidirectional integrable equations. A procedure is presented for deriving multisoliton solutions of one of these equations – a bidirectional Kaup–Kupershmidt equation. These solitons have the unusual property that...

A two-dimensional model for quasigeostrophic flow which exhibits an analogy with the three-dimensional incompressible Euler equations is considered. Numerical experiments show that this model develops sharp fronts without the need to explicitly incorporate any ageostrophic effect. Furthermore, these fronts appear to become singular in finite time. The numerical evidence for singular behavior su...

The behavior of elastic waves in a three-dimensional isotropic incompressible material is studied. Unlike compressible elastodynamics, where there are nonlinear interactions of shear and pressure waves, with incompressible elastodynamics the only waves present are shear waves. In an isotropic system, shear waves are linearly degenerate, and therefore global solutions to the perturbative incompr...