An improved method for solving quasilinear convection diffusion problems on a coarse mesh

A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The Newton-like iterations of the solver are based on the framework of regularized pseudo-transient continuation where the proposed time integrator is a variation on the Newmark strategy, designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite. An algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning. The method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization. Derivation and q-linear local convergence of the method is established, and numerical examples demonstrate the theory including the predicted rate of convergence of the iterations.Comment: 21 pages, 8 figures, 1 tabl

Transient Rayleigh-Benard-Marangoni Convection due to Evaporation : a Linear Non-normal Stability Analysis

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wave number and time. The method is applied to transient Rayleigh-Benard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tension-driven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.Comment: 31 pages, 8 figure

Transient spatiotemporal chaos in the complex Ginzburg-Landau equation on long domains

Numerical simulations of the complex Ginzburg-Landau equation in one spatial dimension on periodic domains with sufficiently large spatial period reveal persistent chaotic dynamics in large parts of parameter space that extend into the Benjamin-Feir stable regime. This situation changes when nonperiodic boundary conditions are imposed, and in the Benjamin-Feir stable regime chaos takes the form of a long-lived transient decaying to a spatially uniform oscillatory state. The lifetime of the transient has Poisson statistics and no domain length is found sufficient for persistent chaos

Numerical solving unsteady space-fractional problems with the square root of an elliptic operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.Comment: 21 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1412.570

Transient convective instabilities in directional solidification

We study the convective instability of the melt during the initial transient in a directional solidification experiment in a vertical configuration. We obtain analytically the dispersion relation, and perform an additional asymptotic expansion for large Rayleigh number that permits a simpler analytical analysis and a better numerical behavior. We find a transient instability, i.e. a regime in which the system destabilizes during the transient whereas the final unperturbed steady state is stable. This could be relevant to growth mode predictions in solidification.Comment: 28 pages, 5 figures. The following article has been accepted for publication in Physics of Fluids. After it is published, it will be found at http://pof.aip.or

Oscillatory convection in binary mixtures: thermodiffusion, solutal buoyancy, and advection

The role of thermodiffusive generation of concentration fluctuations via the Soret effect, their contribution to the buoyancy forces that drive convection, the advective mixing effect of the latter, and the diffusive homogenisation are compared and elucidated for oscillatory convection. Numerically obtained solutions of the field equations in the form of spatially extended relaxed traveling waves, of standing waves, and of the transient growth of standing waves and their transition to traveling waves are discussed as well as spatially localized convective states of traveling waves that are surrounded by the quiescent fluid.Comment: 30 pages, 10 figure