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

Convection-diffusion-reaction models of sedimentation : Numerical approximation, analysis of solutions and inverse problems

The core of this Doctoral thesis is mainly based in the studies of one-dimensional initial-boundary value problems, which are given by a single non-linear hyperbolic partial differential equation (PDE) with non-convex flux function, or by a system of strongly degenerate parabolic PDEs, for the simulation of sedimentation processes of solid particles immersed in a fluid. Particular attention is paid to the case of settling in vessels with varying cross-sectional area. Sedimentation processes are widely used in wastewater treatment (WWT) and mineral processing, where accurate model calibration and reliable simulators are needed. Among the topics covered in the research presented in this thesis are the construction of entropy solutions, the development and implementation of reliable numerical schemes for hyperbolic PDEs (and systems of PDEs), the solution of inverse problems of flux identification, and the disseminationof results to the applied sciences.The outputs of this thesis can be divided into three parts. The first part (Papers I to III) contains the construction of the entropy solutions for the PDE modeling the batch sedimentation in vessels with non-constant cross-sectional area(Paper I and II) and for the PDE modeling centrifugal sedimentation (Paper III). The problem is in both cases solved by the method of characteristics and the types of solutions are distinguished mainly depending on the initial value.Paper II contains the description and solution of the inverse problem of flux identification for the model of sedimentation in conical vessels due to gravity, and Paper III the inverse problem for the model of centrifugal settling. In bothcases, the solution of the inverse problem has the advantage that almost the entire flux function can be identified from only one experiment. These identification methods mean a significant advantage in comparison with the classic one, made by standard tests in cylindrical vessels, in terms of the portion of flux identified. An algorithm necessary for the identification from discrete data is also presented in each problem (Papers II and III).The second part (Papers IV to VI) includes the development of numerical methods for the simulation of sedimentation in WWT. In Paper IV, a numerical scheme for the case of continuous and batch sedimentation in vessels withvarying cross-sectional area is studied. An advantageous CFL condition is derived as an improvement over other numerical methods for the same kind of application. Simulations of continuous and batch settling are also included.Papers V and VI consider reactive settling, where the unknown is a vector of solid and liquid components, and each model is described by a coupled system of convection-diffusion-reaction PDEs. In Paper V, a method-of-lines formulation for the approximation of the model equations is introduced. This formulation has the advantage that it can be solved by any time stepping solver, such as those commonly used in the WWT community where ordinary differentialequations (ODEs) should be solved simultaneously with the PDE system. Additionally, an invariant-region property is proved for the scheme and simulations of interesting scenarios are presented. In Paper VI, sequencing batch reactors (SBRs) are studied. The model equations for the SBRs are derived following Paper V, but with the addition that in this case, the extraction and filling of mixture lead to a moving-boundary problem. The movement of the boundary is described by an ODE which can be precomputed. A reliable numerical scheme that preserves the mass is proposed and numerical simulations for the case of denitrification are shown.The third part (Papers VII and VIII) is related to applications and dissemination of the flux identification methods to the applied sciences. The validation of the inverse problem for batch settling in conical vessels is presented in Pa-per VII. The validation was carried out with data taken from activated sludge collected from the WWT plant in Västerås, Sweden. Paper VIII contains a review of flux identification methods related to PDE models for sedimentation processes. Advantages and disadvantages are discussed, and simulations of identified fluxes with the methods under study are presented.In Chapter 4 the numerical simulation of multidimensional batch sedimentation is discussed and two-dimensional simulations are presented

Simulation of Wastewater Treatment Plants Modeled by a System of Nonlinear Ordinary and Partial Differential Equations

Wastewater treatment consists of mechanical, chemical and biological purification. This master thesis concerns the biological part of the wastewater treatment called the activated sludge process (ASP). Two different mathematical models, one simplified and one complete, of the ASP are investigated. The models contain systems of nonlinear partial and ordinary differential equations. The nonlinearities in the equations give rise to discontinuous solutions, known as shock waves, which complicate the numerical analysis of the equations. The aim of the thesis is to implement the models in MATLAB and investigate how to solve these equations most efficiently with respect to accuracy and speed. Several time discretization schemes including built-in routines in MATLAB will be compared. The results show that a certain semi-implicit method seems to be the most efficient way to solve these equations numerically. Higher order fixed time step methods such as Runge-Kutta methods of order 2 and 4 are not suitable and perform even worse than the very simple Euler method of order 1

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general symmetric diffusion operator of L ́evy type and φ is merely continuous and non-decreasing. We then use this theory to prove con- vergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion op- σ,μ α are the (fractional) Laplacians ∆ and −(−∆)2 for α ∈ (0,2), erators L discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L ́evy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and conver- gence of the methods under minimal assumptions – including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$\partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory.Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysi

ANÁLISIS NUMÉRICO DE UN PROBLEMA INVERSO ORIGINADO EN EL FENÓMENO DE CONTAMINACIÓN AÉREA URBANA

This paper presents the calibration study of a two - dimensional mathematical model for the problem of urban air pollution. It is mainly assumed that air pollution is aected by wind convection, difusion and chemical reactions of pollutants. Consequently, a convection-diusion-reaction equation is obtained as a direct problem. In the inverse problem, the determination of the diusion is analyzed,assuming that one has an observation of the pollutants in a nite time. To solve it numerically the nite volume method is used, the least squares function is considered as cost function and the gradient is calculated with the sensitivity method.En este trabajo se presenta el estudio de calibración de un modelo matemático bidimensional para el problema de contaminación aérea urbana. Se asume principalmente que la contaminación aérea es afectada por la convección del viento, la difusión y las reacciones químicas de los contaminantes.En consecuencia se obtiene, de manera natural, como problema directo una ecuacion de convección-difusión-reacción. En el problema inverso se analiza la determinación de la difusión, asumiendo que se tiene una observación de los contaminantes en un tiempo finito. Para resolverlo numéricamente se utiliza el método de volumenes finitos, se considera como función costo la de mínimos cuadrados y se calcula el gradiente con el método de sensitividad