A series of insoluble heteropolytungstate (H
3PW
12O
40 HPW) salts, Cs
x
H
3−
x
PW
12O
40 (
x
=
0.9
–
3
), were synthesized and characterized using a range of bulk and surface sensitive probes ...including N
2 porosimetry, powder XRD, FTIR, XPS,
31P MAS NMR, and NH
3 calorimetry. Materials with Cs content in the range
x
=
2.0
–
2.7
were composed of dispersed crystallites with surface areas ∼100 m
2 g
−1 and high Brönsted acid strengths
Δ
H
ads
0
(
NH
3
)
=
−
150
kJ
mol
−1
, similar to the parent heteropolyacid. The number of accessible surface acid sites probed by
α-pinene isomerization correlated well with those determined by NH
3 adsorption calorimetry and surface area measurements. Cs
x
H
3−
x
PW
12O
40 were active toward the esterification of palmitic acid and transesterification of tributyrin, important steps in fatty acid and ester processing for biodiesel synthesis. Optimum performance occurs for Cs loadings of
x
=
2.0
–
2.3
, correlating with the accessible surface acid site density. These catalysts were recoverable with no leaching of soluble HPW.
Ureteroscopy is a minimally invasive surgical procedure for the removal of kidney stones. A ureteroscope, containing a hollow, cylindrical working channel, is inserted into the patient's kidney. The ...renal space proximal to the scope tip is irrigated, to clear stone particles and debris, with a saline solution that flows in through the working channel. We consider the fluid dynamics of irrigation fluid within the renal pelvis, resulting from the emerging jet through the working channel and return flow through an access sheath. Representing the renal pelvis as a two-dimensional rectangular cavity, we investigate the effects of flow rate and cavity size on flow structure and subsequent clearance time of debris. Fluid flow is modelled with the steady incompressible Navier–Stokes equations, with an imposed Poiseuille profile at the inlet boundary to model the jet of saline, and zero-stress conditions on the outlets. The resulting flow patterns in the cavity contain multiple vortical structures. We demonstrate the existence of multiple solutions dependent on the Reynolds number of the flow and the aspect ratio of the cavity using complementary numerical simulations and particle image velocimetry experiments. The clearance of an initial debris cloud is simulated via solutions to an advection–diffusion equation and we characterise the effects of the initial position of the debris cloud within the vortical flow and the Péclet number on clearance time. With only weak diffusion, debris that initiates within closed streamlines can become trapped. We discuss a flow manipulation strategy to extract debris from vortices and decrease washout time.
► Analysis of multirate integration methods for multiscale problems. ► Derives computable hybrid a posteriori – a priori error estimates. ► Accounts for component error, projection between scales, ...and finite iteration. ► Includes several examples.
In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. The main purpose of this paper is to present a hybrid a priori –a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations, the use of only a finite number of iterations in the iterative solution of the discrete equations, the numerical error arising in the solution of each component, and the effects on stability arising from the multirate solution. The hybrid estimate has the form of a computable a posteriori leading order expression and a provably-higher order a priori expression. We support this estimate by an a priori convergence analysis. We present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate.
•We develop a multidiscretization method for multiscale parabolic problems.•We derive an accurate a posteriori error estimate.•We account for the effects of finite iteration on the discrete ...solution.•We apply the method to a number of test cases and an interesting application.
This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction–diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators.
Matrix population models have long been used to examine and predict the fate of threatened populations. However, the majority of these efforts concentrate on long-term equilibrium dynamics of linear ...systems and their underlying assumptions and, therefore, omit the analysis of transience. Since management decisions are typically concerned with the short term (<100 years), asymptotic analyses could lead to inaccurate conclusions or, worse yet, critical parameters or processes of ecological concern may go undetected altogether.
We present a stage-structured, deterministic, nonlinear, disease model which is parameterized for the population dynamics of high-elevation white pines in the face of infection with white pine blister rust (WPBR). We evaluate the model using newly developed software to calculate sensitivity and elasticity for nonlinear population models at any projected time step. We concentrate on two points in time, during transience and at equilibrium, and under two scenarios: a regenerating pine stand following environmental disturbance and a stand perturbed by the introduction of WPBR.
The model includes strong density-dependent effects on population dynamics, particularly on seedling recruitment, and results in a structure favoring large trees. However, the introduction of WPBR and its associated disease-induced mortality alters stand structure in favor of smaller stages. Populations with infection probability (β) 0.1 do not reach a stable coexisting equilibrium and deterministically approach extinction.
The model enables field observations of low infection prevalence among pine seedlings to be reinterpreted as resulting from disease-induced mortality and short residence time in the seedling stage.
Sensitivities and elasticities, combined with model output, suggest that future efforts should focus on improving estimates of within-stand competition, infection probability, and infection cost to survivorship. Mitigating these effects where intervention is possible is expected to produce the greatest effect on population dynamics over a typical management timeframe.
In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute ...accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.
We consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction–diffusion partial differential equations coupled to a microscale ...model comprising a system of nonlinear ordinary differential equations. A motivating example is modeling the electrical activity of the heart taking into account the chemistry inside cells in the heart. Such multiscale models are computationally challenging due to the multiple scales in time and space that are involved.
We describe a mathematically consistent approach to couple the microscale and macroscale models based on introducing an intermediate “coupling scale”. Since the ordinary differential equations are defined on a much finer spatial scale than the finite element discretization for the partial differential equation, we introduce a Monte Carlo approach to sampling the fine scale ordinary differential equations. We derive goal-oriented a posteriori error estimates for quantities of interest computed from the solution of the multiscale model using adjoint problems and computable residuals. We distinguish the errors in time and space for the partial differential equation and the ordinary differential equations separately and include errors due to the transfer of the solutions between the equations. The estimate also includes terms reflecting the sampling of the microscale model. Based on the accurate error estimates, we devise an adaptive solution method using a “blockwise” approach. The method and estimates are illustrated using a realistic problem.
•Stochastic mesoscale coupling mechanism for multiscale model.•A posteriori error estimate for multiscale coupled system.•Adaptive error control for multiscale model.•Multiscale model for heart.
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