On diffusively corrected multispecies kinematic flow models

This presentation provides a survey of some recent results related to efficient numerical methods for the numerical solution of a class of convection-diffusion systems that arise as one-dimensional models of the flow of one (disperse) substance through a continuous fluid. Applications include the settling of polydisperse suspensions of solid particles in a viscous fluid, multiclass vehicular traffic under the effect of anticipation distances and reaction times, the settling of dipersions and emulsions, and chromatography. In many of these applications the system becomes strongly degenerate. For the numerical solution, this fact poses a number of difficulties whose partial solution will be addressed. For instance, it is well known that implicit-explicit (IMEX) numerical scheme that are based on discretizing the convective and diffusive parts are a potentially suitable tool to avoid the severe time step limitation associated with fully explicit discretization. However, their implementation relies on the efficient numerical solution of the nonlinear systems of algebraic equations arising from the discretization which can not be achieved by standard Newton-Raphson techniques when the diffusion coefficients are discontinuous. A combined smoothing and line search technique solves the problem of solving the corresponding nonlinearly implicit equations. Alternatively, this problem can be avoided by the construction of so-called linearly implicit methods that are slightly less accurate, but noticeably more efficient than their nonlinearly implicit counterparts. The main collaborators in this research are Pep Mulet (Universitat de Valencia, Spain) and Luis Miguel Villada (Universidad del Bío-Bío, Concepción, Chile).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure

On boundary conditions for multidimensional sedimentation-consolidation processes in closed vessels

The two-phase flow of a flocculated suspension in a closed settling vessel with inclined walls is investigated within the phenomenological theory of sedimentation-consolidation processes. We formulate possible wall boundary conditions and use these conditions to derive spatially one-dimensional field equations for planar flows and flows which are symmetric with respect to the vertical axis. For both kinds of flows we assume a general geometry of the sedimentation vessel and include the study of a compressible sediment layer. We analyze the special cases of a conical vessel, a roof-shaped vessel and a vessel with parallel inclined walls. The case of a small initial time and a large time for the final consolidation state leads to explicit expressions for the flow fields. From a mathematical point of view, the resulting initial-boundary value problems are well posed and can be solved numerically by a simple adaptation of one of the newly developed numerical schemes for strongly degenerate convection-diffusion problems. However, from a physical point of view, both the analytical and numerical results rise doubts concerning the validity of the general field equations. In particular, the strongly reduced form of the linear momentum balance seems to be an oversimplification. Included in our discussion as a special case are the Kynch theory and well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit similar shortcomings

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.Comment: 27 pages, 14 figure

Adaptive Multiresolution Methods for the Simulation ofWaves in Excitable Media

We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley's approach (Barkley in Phys. D 49:61-70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform gri

Towards improved 1-D settler modelling : calibration of the Bürger model and case study

Recently, Burger et al. (2011) developed a new 1-D SST model which allows for more realistic predictions of the sludge settling behaviour than traditional 1-D models used to date. However, the addition of a compression function in this new 1-D model complicates the model calibration. This study aims to report advances in the calibration of this novel 1-D model. Data of the evolution of the sludge blanket height during batch settling experiments were collected at different initial solids concentrations. Based on the linear slopes of the batch settling curves the hindered settling velocity functions by Vesilind (1968) and Takacs et al. (1991) were calibrated. Although both settling velocity functions gave a good fit to the experimental data, very large confidence intervals were found for the parameters of the settling velocity by Takacs. Global sensitivity analysis showed that it is not possible to find a unique set of parameter values for the settling function by Takacs based on experimental data of the hindered settling velocity. Subsequently, the calibrated Vesilind settling velocity was implemented in the 1-D model by Burger et al. (2011) and the parameters of the additional compression function were calibrated by fitting the model by Burger et al. (2011) to the batch settling curves. Simulation results showed that while the 1-D model by Takacs et al. (1991) underpredicted the experimental data of sludge blanket heights, the model by Burger et al. (2011) was able to predict the experimental data far more accurately. However, a global sensitivity analysis showed that no unique optimum for the combined set of hindered and compression parameters could be found