A multiscale model for self-assembly with secondary nucleation-like properties

Most of the biological polymers that make up our cells and tissues are hierarchically structured. For biopolymers ranging from collagen, to actin, to fibrin, this hierarchy provides vitally important versatility, allowing a multitude of structurally and functionally distinct structures to be constructed from a limited set of biomolecular constituents. This structural hierarchy must be encoded in the self-assembly process, from the earliest stages onward, in order to produce the appropriate substructures in the correct sequence. In this Letter, we explore the kinetics of such multi-stage self-assembly processes in a model system which is formulated as a set of discrete master equations capturing the underlying hierarchical molecular-scale process, but which may be homogenized to yield a practical, continuum description in terms of PDEs to compare to bulk experiments such as light scattering or turbidity measurements. We present the general framework, and apply it to recent turbidimetry data on the self-assembly of collagen fibrils. Furthermore, our analysis suggests a connection between diffusion-limited aggregation kinetics and fibril growth, supported by slow, power-law growth at very long timescales observed in both systems

Embedded WENO : a design method to improve existing WENO schemes

Embedded WENO methods utilize all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes underuse this possibility close to large gradients or discontinuities. 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. The embedded methods are demonstrated to be improvements over their standard counterparts by several numerical examples. All the embedded methods presented have virtually no added computational effort compared to their standard counterparts. Keywords: Essentially non-oscillatory, WENO, high-resolution scheme, hyperbolic conservation laws, nonlinear interpolation, spectral analysis

Full linear multistep methods as root-finders

\u3cp\u3eRoot-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brent's method that is guaranteed to converge.\u3c/p\u3

High-order embedded WENO schemes

\u3cp\u3eEmbedded WENO schemes are a new family of weighted essentially nonoscillatory schemes that always utilise all adjacent smooth substencils. This results in increased control over the convex combination of lower-order interpolations. We show that more conventional WENO schemes, such as WENO-JS and WENO-Z (Borges et al., J. Comput. Phys., 2008; Jiang and Shu, J. Comput. Phys., 1996), do not exhibit this feature and as such do not always provide a desirable linear combination of smooth substencils. In a previous work, we have already developed the theory and machinery needed to construct embedded WENO methods and shown some five-point schemes (van Lith et al., J. Comput. Phys., 2016). Here, we construct a seven-point scheme and show that it too performs well using some numerical examples from the one-dimensional Euler equations.\u3c/p\u3

Structural adjustment A general overview, 1980-1989

SIGLEAvailable from British Library Document Supply Centre-DSC:4362.5299(MU-IDPM-DP--21) / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Active flux schemes on moving meshes with applications to geometric optics

\u3cp\u3eActive flux schemes are finite volume schemes that keep track of both point values and averages. The point values are updated using a semi-Lagrangian step, making active flux schemes highly suitable for geometric optics problems on phase space, i.e., to solve Liouville's equation. We use a semi-discrete version of the active flux scheme. Curved optics lead to moving boundaries in phase space. Therefore, we introduce a novel way of defining the active flux scheme on moving meshes. We show, using scaling arguments as well as numerical experiments, that our scheme outperforms the current industry standard, ray tracing. It has higher accuracy as well as a more favourable time scaling. The numerical experiments demonstrate that the active flux scheme is orders of magnitude more accurate and faster than ray tracing.\u3c/p\u3

A novel scheme for Liouville's equation with a discontinuous Hamiltonian and applications to geometrical optics

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

Existence and uniqueness of solutions to Liouville's equation and the associated flow for Hamiltonians of bounded variation

We prove existence and uniqueness for solutions to Liouville's equation for Hamiltonians of bounded variation. These solutions can be interpreted as the limit of a sequence generated by a series of smooth approximations to the Hamiltonian. This results in a converging sequence of approximations of solutions to Liouville's equation. As an added perk, our method allows us to prove a generalisation of Liouville's theorem for Hamiltonians of bounded variation. Furthermore, we prove there exists a unique flow solution to the Hamilton equations and show how this can be used to construct a solution to Liouville's equation. Key words: partial differential equations, geometrical optics, Liouville's equation, flow