Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

The generic structure of solutions of initial value problems of hyperbolic- elliptic systems, also called mixed systems, of conservation laws is not yet fully understood. One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations, so-called oscillatory waves, which are (in general, measure-valued) solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region. To capture these solutions, usually a fine computational grid is required. In this work, a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type. The hyperbolic-elliptic 2×2 systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension. In the latter case, varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space, giving rise to different kinds of oscillation waves

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Comparative Assessment of Adaptive-Stencil Finite Difference Schemes for Hyperbolic Equations with Jump Discontinuities

High-fidelity numerical solution of hyperbolic differential equations for functions with jump discontinuities presents a particular challenge. In general, fixed-stencil high-order numerical methods are unstable at discontinuities, resulting in exponential temporal growth of dispersive errors (Gibbs phenomena). Schemes utilizing adaptive stencils have shown to be effective in simultaneously providing high-order accuracy and long-time stability. In this Thesis, the elementary formulation of adaptive-stenciling is described in the finite difference context. Basic formulations are provided for three adaptive-stenciling methods: essentially non-oscillatory (ENO), weighted essentially non-oscillatory (WENO), and energy-stable weighted essentially non-oscillatory (ESWENO) schemes. Examples are presented to display some of the relevant properties of these schemes in solving one-dimensional and two-dimensional linear and nonlinear hyperbolic differential equations with discontinuities

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

On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

Finite element and finite difference discretizations for evolutionary convection-dif\-fusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods

On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods

Hiperbolicidad de un modelo de sedimentación polidispersa e información característica para la implementación del método WENO espectral con multirresolución adaptativa

En este trabajo se desarrollaron de manera muy detallada los cálculos para mostrar la hiperbolicidad del modelo de sedimentación de suspensiones polidispersas de Höfler - Schwarzer. Como consecuencia del análisis del carácter hiperbólico del modelo se obtiene la información característica necesaria para la implementación de un esquema numérico de alto orden teniendo en cuenta las ventajas que implica el uso de esquemas espectrales en comparación con esquemas por componentes. Concretamente, implementamos métodos de multirresolución adaptativa para el popular esquema numérico de alto orden WENO5 espectral propuesta por Harten. Se da evidencia numérica de las ventajas al utilizar un esquema de alto orden con información espectral y combinado con el método adaptativo de multirresolución para un problema descrito por un sistema de leyes de conservación hiperbólico medidos por compresión de datos.MaestríaMagister en Matemática

Hiperbolicidad e implementación del método WENO espectral con multiresolución adaptativa para el modelo tráfico de Lighthill-Whitham-Richards

En este trabajo se desarrollaron en forma detallada los cálculos para mostrar la hiperbolicidad del modelo multiclase de tráfico vehicular de Lighthill-Whitham-Richards. Como subproducto del análisis del carácter hiperbólico del modelo se obtiene la información característica necesaria para la implementación de un esquema numérico de alto orden teniendo en cuenta las ventajas que implica el uso de esquemas espectrales en comparación con esquemas por componentes. Más concretamente, se usó el popular esquema WENO de orden cinco combinado con una estrategia adaptativa de multiresolución propuesta por Harten. Se da evidencia numérica de las ventajas al utilizar un esquema de alto orden con información espectral y combinado con el método adaptativo de multiresolución para un problema descrito por un sistema de leyes de conservación hiperbólico.MaestríaMagister en Matemática