Interface tracking methods and gradient-augmented algorithms: theory and applications

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The objective of the proposed research is the development of a new class of methods for the solution of Partial Differential Equations (PDEs) with interface conditions and their application to problems in engineering and computational science. Since it is as important to represent and evolve the interface accurately as it is to correctly discretize operators in order to develop numerical methods with predictive capabilities, our approach will be three-pronged. First, the representation and tracking of small structures using level set methods is investigated. Level set methods evolve a surface or a curve using a level set function defined on an Eulerian grid. Commonly used approaches suffer from a loss of mass. This investigation proposes to remedy the above-mentioned problems by incorporating gradient information into the calculation, by providing additional equations. The knowledge of evolved gradient information does not allow an actual simulation of some process on a sub-grid scale, but it will allow to capture and to track structures smaller than the grid size. Second, the gradient-augmented approach may improve the accuracy in calculating differential quantities where gradients (or higher derivatives) play a role. We will investigate the extension of the proposed gradient-augmented "philosophy" to other class of PDEs such as Hamilton-Jacobi equations, Poisson equation with interface jump conditions and the two-phase incompressible Navier-Stokes Equations. The long-term aim of the proposed investigations is to develop a full theory of gradient-augmented schemes, and to provide a set of tools and methods for the solution of PDEs with interface. Third, many application to various problems involving interface conditions or discontinuities will be investigated. We propose applying the methods to study incompressible two-phase Navier-Stokes equations in various situations, including complex fluids. We will investigate applicability of the gradient-augmented framework to solid-fluid interaction problems, and shocks in traffic flows as prototype non-linear conservation laws with great practical interest.