Sparse Matrices as a Practical Way to Solve Systems of Partial Differential Equations when the Boundary Conditions are in Opposite Locations

• Tipo: Artigo em Livro de Atas de Conferência Nacional
• Data: 1998

Autores

António Fiúza (Autor) FEUP

Ata de Conferência Nacional

Título: 2nd Meeting on Numerical Methods for Differential Equations

Páginas: 490-500

2nd Meeting on Numerical Methods for Differential Equations

Departamento de Matemática, Universidade de Coimbra, 25 a 27 de Fevereiro de 1998

Outras Informações

Abstract (EN): A high number of technological operations involve mass transfer phenomena between two phases that flow counter-currently. These systems can be described by a system of partial differential equations where the response variables are variable in space and time. It is an usual procedure to use numerical methods and among these the finite differences are a common and generalized alternative of easy implementation. But as we are dealing with a counter-current operation the boundary conditions are defined in the extreme opposite sides of the spatial coordinates. This situation requires the simultaneous usage of descending differences for one of the response variables and ascending differences for the other. This particular set of boundary conditions usually denies the possibility of finding a recursive numerical strategy where solutions irradiate from the solving of the discretized system in each space-time cell, to the neighbouring cells. Under this conditions the numerical solution requires to solve a double system of algebraic equations for each space-time cell, which implies to solve a simultaneous system of thousands of algebraic equations to thousand of unknown variables. Even with large increments in space and time discretizations, the resulting systems are enormous and can not be solved due to restrictions in memory in ordinary personal computers. Another peculiarity of the resulting system is the high number of zero elements in the system matrix. Under these circumstances the usage of sparse matrices allows the analytical resolution of the discretized system due to the high economy in memory generated. Simultaneously the time required for the calculations, which could be of several hours, suffers a shortage to the order of seconds, once the required amount of time is proportional to the number of arithmetic operations on nonzero quantities. This methodology is applied to a paradigmatic distribute parameter model of a technological operation.

Idioma: Português

Tipo (Avaliação Docente): Científica

Nº de páginas: 11