Discretization schemes that preserve non-trivial symmetry groups of partial differential equations have received increasing attention over the past 20 years. These investigations parallel ongoing efforts to construct conservative discretizations and belong to the field of geometric numerical integration. It has been found that preserving symmetry groups for evolution equations is in general not possible on stationary discretization meshes. The problem of invariant discretization is therefore intimately linked to the problem of grid adaptation. In this talk we will introduce different strategies that allow constructing discretization schemes with symmetry properties on r-adaptive meshes. The proposed strategies rely on the use of invariant evolution–projection techniques and on invariant meshless discretizations. We will illustrate the different approaches by carrying out invariant numerical integrations for several evolution equations with relevance in geophysical fluid dynamics.

Phagocytosis is one of endocytosis functions to capture vesicles or microorganism into a cell. The lipid binding mechanism between vesicles and membranes plays a crucial role in the cell engulf process, most literatures focused on the discussion of membrane equilibrium state or static problems without the consideration of fluid flow. In this talk, we present a mathematical model in the immersed boundary (IB) formulation which consists of fluid equations, fluid-structure interaction, and molecule concentration equations, to study the role of interfacial properties (surface tension and bending rigidity) in phagocytosis.

A two-layer structure is used for the membrane-bound receptor-ligand binding process, and interfacial forces are derived from Canham-Helfrich Hamiltonian. The membrane-bound molecules are treated as insoluble surfactants such that the molecules after binding are regarded as a product after chemical reaction. Mass conservation constraint and the law of mass action are imposed to obtain the corresponding governing equations of the membrane-bound molecules. A conventional IB method and a conservative numerical scheme with the consideration of reaction for the membrane molecules are proposed. In numerical experiments, we give an energy balance investigation to conform the numerical scheme, then we examine the engulf process by altering the surface energy reduction, the ratio of bending rigidity, vesicle size, and an adhesion energy assistance. The studies of convection-diffusion-reaction effect to engulf process are also presented.

This is a joint work with Y-H Tseng.

We use a recent geodesic theory of transport barriers to compute parametrized surfaces acting as barriers to transport in three-dimensional steady flows. These barriers are formed by shear-type Lagrangian Coherent Structures (LCS) that approximate two-dimensional invariant tori. Our approach uncovers barrier surfaces at a previously unseen level of detail and accuracy. As applications, we discuss integrable and non-integrable three-dimensional fluid flows.

The finite-time Lyapunov exponent (FTLE) is extensively used as a criterion to reveal fluid flow structures, including unsteady separation/attachment surfaces and vortices, in laminar and turbulent flows. However, for large and complex problems, flow structure identication demands computational methodologies that are more accurate and effective. With this objective in mind, we propose a new set of ordinary differential equations to compute the flow map, along with its first (gradient) and second order (Hessian) spatial derivatives. We show empirically that the gradient of the flow map computed in this way improves the pointwise accuracy of the FTLE field. Furthermore, the Hessian allows for simple interpolation error estimation of the flow map, and the construction of a continuous optimal and multiscale $L_p$ metric. The Lagrangian particles, or nodes, are then iteratively adapted on the flow structures revealed by this metric. It is found that Lagrangian Coherent Structures are best revealed with the minimum number of vertices with the $L_1$ metric.