Developing improved time stepping schemes for nonlinear phase change models

Phase change problems occur in a large number of practical applications and processes from the food industry through to heavy engineering. In particular, many industrial processes involve the solidification of molten metals, for example the highly controlled casting of components such as aircraft wings or turbine blades. The physical properties of the resulting cast products depend strongly on the fine detail of the solidification itself – hence there is enormous interest in understanding solidification processes in great detail.

There are a range of mathematical models that aim to describe phase change problems such as solidification: the most popular classes are so-called phase-field methods. In these models the interface is captured through the use of a phase variable which varies continuously (but sharply) from 0 to 1 (say) between the two phases. The resulting system of partial differential equations is nonlinear and parabolic in nature. Explicit time-stepping is generally not feasible due to stability constraints and more sophisticated approaches are therefore required.

At Leeds we have developed new computational tools to solve these systems using conventional implicit time-stepping schemes coupled with a novel nonlinear multigrid method for the spatial part of the discretization. This project will improve upon this existing software though the development and implementation of a new algorithm for time stepping. This will be based upon a novel technique based on spectral deferred corrections.

Once the new technique has been developed and integrated into to the code we will explore its performance against the existing backward differentiation formula. One of the great advantages of the proposed approach is that it will provide the opportunity to implement new parallel algorithms too – permitting parallelism in time as well as (the more conventional) spatial parallelism.