a b s t r a c tIn this paper we present and compare two unconditionally energy stable finite-difference schemes for the phase field crystal equation. The first is a one-step scheme based on a convex splitting of a discrete energy by Wise et al. [S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press]. In this scheme, which is first order in time and second order in space, the discrete energy is non-increasing for any time step. The second scheme we consider is a new, fully second-order two-step algorithm. In the new scheme, the discrete energy is bounded by its initial value for any time step. In both methods, the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable, regardless of time step-size. We solve the nonlinear equations using an efficient nonlinear multigrid method. Numerical simulations are presented and confirm the stability, efficiency and accuracy of the schemes.

Recent advances in the science and technology of composites utilizing carbon nanostructures are reviewed, including experimental results and modelling studies of composite properties and processing. Carbon nanotubes are emphasized, with other carbon nanostructures such as fullerenes, ultradispersed diamond clusters and diamond nanorods also being discussed.

In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In * Currently at Haerbin Inst. Techn., China 1 arXiv:1208.2643v1 [math.NA] 8 Aug 2012 particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.which is significantly different from schemes we propose and analyze. In particular, theirs is not expected to be unconditionally solvable or energy stable. The authors did, however, give some evidence of the efficiency of their multigrid solver. We will conduct a similar study for our multigrid solver in this paper. The MPFC scheme in [10] is more or less the same as the first-order convex-splitting that we devised earlier in [18,19]. The first-order convex splitting scheme in our work [18,19] has two fundamental properties. It is unconditionally energy stable and unconditionally uniquely solvable. We rigorously analyzed this convex splitting scheme in [18,19], but we did not provide a practical solution strategy. That gap is filled here.In [8,21] we presented first and second-order accurate (in time) finite difference schemes for the purely parabolic PFC model, based on a convex splitting framework applied to the physical energy. The convexity splitting idea -in the context of first order (in time) convex splitting schemes -is generally credited to [7]. One of the main advances in [8] -and the more recent work in [14] -was the demonstration that the convex splitting framework can be extended to second-order (in time) methods as well, in a natural way. While the motivation in [8] was the application to the PFC model, the second-order convex splitting idea that we developed is, in fact, rather general.The main goal of this paper is to apply the first and second-order convex splitting framework to the MPFC equation. The principal challenge in doing so is that the extension to damped wave dynamics appears, at first sight, not so straightforward. This obstacle was first surmounted in [18,19], as already mentioned, where we extended the first-order convexity splitting idea for the MPFC model. Specifically, we showed the existence of a pseudo energy (different from the physical energy),...

Ethanol can be produced via an intracellular photosynthetic process in cyanobacteria (blue-green algae), excreted through the cell walls, collected from closed photobioreactors as a dilute ethanol-in-water solution, and purified to fuel grade ethanol. This sequence forms the basis for a biofuel production process that is currently being examined for its commercial potential. In this paper, we calculate the life cycle energy and greenhouse gas emissions for three different system scenarios for this proposed ethanol production process, using process simulations and thermodynamic calculations. The energy required for ethanol separation increases rapidly for low initial concentrations of ethanol, and, unlike other biofuel systems, there is little waste biomass available to provide process heat and electricity to offset those energy requirements. The ethanol purification process is a major consumer of energy and a significant contributor to the carbon footprint. With a lead scenario based on a natural-gas-fueled combined heat and power system to provide process electricity and extra heat and conservative assumptions around the ethanol separation process, the net life cycle energy consumption, excluding photosynthesis, ranges from 0.55 MJ/MJ(EtOH) down to 0.20 MJ/ MJ(EtOH), and the net life cycle greenhouse gas emissions range from 29.8 g CO₂e/MJ(EtOH) down to 12.3 g CO₂e/MJ(EtOH) for initial ethanol concentrations from 0.5 wt % to 5 wt %. In comparison to gasoline, these predicted values represent 67% and 87% reductions in the carbon footprint for this ethanol fuel on a energy equivalent basis. Energy consumption and greenhouse gas emissions can be further reduced via employment of higher efficiency heat exchangers in ethanol purification and/ or with use of solar thermal for some of the process heat.

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