Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the TTLEM 1.0 model

Landscape evolution models (LEMs) allow the study of earth surface responses to changing climatic and tectonic forcings. While much effort has been devoted to the development of LEMs that simulate a wide range of processes, the numerical accuracy of these models has received less attention. Most LEMs use first-order accurate numerical methods that suffer from substantial numerical diffusion. Numerical diffusion particularly affects the solution of the advection equation and thus the simulation of retreating landforms such as cliffs and river knickpoints. This has potential consequences for the integrated response of the simulated landscape. Here we test a higher-order flux-limiting finite volume method that is total variation diminishing (TVD-FVM) to solve the partial differential equations of river incision and tectonic displacement. We show that using the TVD-FVM to simulate river incision significantly influences the evolution of simulated landscapes and the spatial and temporal variability of catchment-wide erosion rates. Furthermore, a two-dimensional TVD-FVM accurately simulates the evolution of landscapes affected by lateral tectonic displacement, a process whose simulation was hitherto largely limited to LEMs with flexible spatial discretization. We implement the scheme in TTLEM (TopoToolbox Landscape Evolution Model), a spatially explicit, raster-based LEM for the study of fluvially eroding landscapes in TopoToolbox 2.

Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the TTLEM 1.0 model

Landscape evolution models (LEM) allow studying the earth surface response to a changing climatic and tectonic forcing. While much effort has been devoted to the development of LEMs that simulate a wide range of processes, the numerical accuracy of these models has received much less attention. Most LEMs use first order accurate numerical methods that suffer from substantial numerical diffusion. Numerical diffusion particularly affects the solution of the advection equation and thus the simulation of retreating landforms such as cliffs and river knickpoints with potential unquantified consequences for the integrated response of the simulated landscape. Here we present TTLEM, a spatially explicit, raster based LEM for the study of fluvially eroding landscapes in TopoToolbox 2. TTLEM prevents numerical diffusion by implementing a higher order flux limiting total volume method that is total variation diminishing (TVD-TVM) and solves the partial differential equations of river incision and tectonic displacement. We show that the choice of the TVD-TVM to simulate river incision significantly influences the evolution of simulated landscapes and the spatial and temporal variability of catchment wide erosion rates. Furthermore, a 2D TVD-TVM accurately simulates the evolution of landscapes affected by lateral tectonic displacement, a process whose simulation is hitherto largely limited to LEMs with flexible spatial discretization. By providing accurate numerical schemes on rectangular grids, TTLEM is a widely accessible LEM that is compatible with GIS analysis functions from the TopoToolbox interface.

Mathematical Model and Fabrication of Multi-Layer Electrochemical Glucose Sensors

The design and fabrication of multi-layer amperometric electrochemical glucose sensors is dependent upon the diffusional kinetics of the chemical/biochemical species which contribute to the sensor’s response. Considerable effort has been carried out to coat the working electrode with appropriate glucose flux-limiting membranes which is pertinent for superior in vivo performance, and hence requires a careful understanding of the participating species within the sensor cross-sectional architecture. This contribution reports the computational modeling of Clark’s first generation amperometric glucose sensor coated with an electro-polymerized glucose oxidase (GOx) layer along with a layer of polyurethane (PU) employed to reduce the glucose-influx in order to generate linear operation over the normal physiological glucose range in vivo. The model was programmed using MATLAB and utilizes the finite-difference method for the solution to the enzymatic reaction-based diffusion equations. Additionally, experimental devices were fabricated, tested and compared with the simulated results. The simulation of these devices have been shown to align well with experimentally fabricated devices in terms of amperometric current density. The increase in device linearity with the addition of the outer glucose-flux limiting PU membrane corroborate our experimental findings reported in this study which can be used as a powerful analytical tool in designing high–performance next generation implantable glucose sensors.

Restoring mass conservation to shallow ice flow models over complex terrain

Numerical simulation of glacier dynamics in mountainous regions using zero-order, shallow ice models is desirable for computational efficiency so as to allow broad coverage. However, these models present several difficulties when applied to complex terrain. One such problem arises where steep terrain can spuriously lead to large ice fluxes that remove more mass from a grid cell than it originally contains, leading to mass conservation being violated. This paper describes a vertically integrated, shallow ice model using a second-order flux-limiting spatial discretization scheme that enforces mass conservation. An exact solution to ice flow over a bedrock step is derived for a given mass balance forcing as a benchmark to evaluate the model performance in such a difficult setting. This benchmark should serve as a useful test for modellers interested in simulating glaciers over complex terrain.

A parameter-free smoothness indicator for high-resolution finite element schemes

This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.