Maximum norm error estimates of efficient difference schemes for second-order wave equations

AbstractThe three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above

Models of Delay Differential Equations

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin

Interpolação polinomial com multiextrapolação Richardson para reduzir o erro de discretização em malhas não uniformes 1D

Orientador : Prof. Dr. Carlos Henrique MarchiCoorientador : Prof. Dr. Márcio André MartinsDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 30/03/2015Inclui referências : f. 78-80Resumo: Neste trabalho emprega-se a Multiextrapolação de Richardson (MER) para reduzir o erro de discretização (Eh), na área de Dinâmica dos Fluidos Computacional (em inglês Computational Fluid Dynamics (CFD)), que consiste em uma técnica de pós processamento de dados para reduzir Eh. Para atingir este objetivo utiliza-se uma nova metodologia, proposta em 2013 para malhas uniformes, em que se aplica interpolações polinomiais. Através desta metodologia, MER teve seu desempenho melhorado: a magnitude dos erros de discretização reduziu progressivamente com o refinamento da malha, com um concomitante aumento das suas ordens de acurácia, até mesmo em variáveis de interesse cuja localização não é fixa, até então na literatura, MER era considerada de baixo desempenho neste tipo de variável. Com o intuito de estender este significativo resultado de 2013, esta metodologia foi estendida para malhas não uniformes unidimensionais. Como problema-modelo é considerado a equação de Poisson 1D, utilizando dois tipos de malha inicial, e refinamento uniforme. A discretização dessa equação foi realizada com o método de Diferenças Finitas. Nas variáveis de interesse estudadas, testou-se vários graus de interpolação e foram alcançados resultados semelhantes com os apresentados para este mesmo problema utilizando malhas uniformes, com MER, isto é, o erro de discretização teve redução significativa. Palavras-chave: Erro de discretização. Multiextrapolação de Richardson (MER). Interpolação polinomial. Dinâmica dos fluidos computacional (CFD). Equação de Poisson 1D. Método de Diferenças Finitas.Abstract: This work applies the Repeated Richardson Extrapolation (RRE) to reduce the discretization error (Eh) in Computational Fluid Dynamics (CFD), consisting of a post-data processing technique to reduce Eh. Therefore, to use RRE, we use a new methodology, created in 2013 for uniform grids, which works with the use of polynomial interpolation. Using this methodology, RRE had its performance improved: the magnitude of the discretization errors was reduced progressively with mesh refinement, with a concomitant increase in its accuracy orders, even variables of interest whose location is not fixed, so far in the literature, RRE was considered underperforming in this type of variable. We adapted this methodology to non-uniform one-dimensional meshes, with intent of extending this significant recent result for non-uniform grids. As problem-model we consider the Poisson equation 1D using two types of initial mesh with uniform refinement. The discretization of equation is performed using the Finite Difference Method. In the interest variables studied in this work, it was tested various degrees of interpolation and obtained similar results to those presented to the same problem for uniform meshes with RRE, in the other words, the Eh had a significant reduction. Keywords: Discretization error, Repeated Richardson Extrapolation (RRE), Polynomial Interpolation, Computational Fluid Dynamics (CFD), 1D Poisson Equation, Finite Difference Method