Embedded WENO methods utilize all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu [1] and on the WENO-Z scheme of Borges et al.[2] are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.

A novel scheme is developed that computes numerical solutions of Liouville's equation with a discontinuous Hamiltonian. It is assumed that the underlying Hamiltonian system has well-defined behaviour even when the Hamiltonian is discontinuous. In the case of geometrical optics such a discontinuity yields the familiar Snell's law or the law of specular reflection. Solutions to Liouville's equation should be constant along curves defined by the Hamiltonian system when the right-hand side is zero, i.e., no absorption or collisions. This consideration allows us to derive a new jump condition, enabling us to construct a first-order accurate scheme. Essentially, the correct physics is built into the solver. The scheme is tested in a two-dimensional optical setting with two test cases, the first using a single jump in the refractive index and the second a compound parabolic concentrator. For these two situations, the scheme outperforms the more conventional method of Monte Carlo ray tracing.

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Most of the biological polymers that make up our cells and tissues are hierarchically structured. For biopolymers ranging from collagen, to actin, to fibrin and amyloid fibrils this hierarchy provides vitally important versatility. The structural hierarchy must be encoded in the self-assembly process, from the earliest stages onward, in order to produce the appropriate substructures. In this letter, we explore the kinetics of multistage self-assembly processes in a model system which allows comparison to bulk probes such as light scattering. We apply our model to recent turbidimetry data on the self-assembly of collagen fibrils. Our analysis suggests a connection between diffusion-limited aggregation kinetics and fibril growth, supported by slow, power-law growth at very long time scales.

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