Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of the noise with a semi--implicit Euler--Maruyama method in time and in space we analyse a finite element method (although extension to finite differences or finite volumes would be possible). We prove convergence in the root mean square $L^{2}$ norm for a diffusion reaction equation and diffusion advection reaction equation. We present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation. We see from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi--implicit Euler--Maruyama method

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a \textit{finite} transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time (ii) mean plus one standard deviation of action time and (iii) a new approach derived by approximating the large time asymptotic behaviour of the cumulative distribution function. The new approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance $\delta$ and the $(k-1)$th and $k$th moments ($k \geq 1$) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. Firstly, while the first two approaches are useful at characterising the time scale of the transition, they do not provide accurate estimates for diffusion processes. Secondly, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits, using only the moments, with the accuracy increasing as the index $k$ is increased.Comment: 17 pages, 2 figures, accepted version of paper published in Physical Review

A second order scheme for a Robin boundary condition in random walk algorithms

Random Walk (RW) is a common numerical tool for modeling the Advection-Diffusion equation. In this work, we develop a second order scheme for incorporating a heterogeneous reaction (i.e., a Robin boundary condition) in the RW model. In addition, we apply the approach in two test cases. We compare the second order scheme with the first order one as well as with analytical and other numerical solution. We show that the new scheme can reduce the computational error significantly, relative to the first order scheme. This reduction comes at no additional computational cost

Spatial Dynamic Models for Fishery Management and Waterborne Disease Control

As the human population continues to grow, there is a need for better management of our natural resources in order for our planet to be able to produce enough to sustain us. One important resource we must consider is marine fish populations. We use the tool of optimal control to investigate harvesting strategies for maximizing yield of a fish population in a heterogeneous, finite domain. We determine whether these solutions include no-take marine reserves as part of the optimal solution. The fishery stock is modeled using a nonlinear, parabolic partial differential equation with logistic growth, movement by diffusion and advection, and with Robin boundary conditions. The objective for the problem is to find the harvest rate that maximizes the discounted yield. Optimal harvesting strategies are found numerically. Infectious diseases are another area of concern for the human population. Recently, questions have been raised as to the importance of spatial features on disease spread and how movement patterns affect management strategies. The role of spatial arrangements in a metapopulation on the spread and management strategies of a cholera epidemic is investigated. We consider how the movement of individuals and water affects the optimal vaccination strategy. For each metapopulation, the model has an Susceptible-Infected-Recovered (SIR) system of differential equations coupled with an equation modeling the concentration of Vibrio cholerae in an aquatic reservoir. The model is used to compare spatial arrangements and varying scenarios to draw conclusions on how to effectively manage outbreaks. The work is motivated by the recent cholera outbreak in Haiti

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting