Low-ram algorithm for solving 3-d natural convection problems using orthogonal collocation

Algoritmo de bajo consumo de ram para resolver problemas de convección natural 3-d, usando colocación ortogonal

H. Jiménez-Islas¹*, M. Calderón-Ramírez³, F.I. Molina-Herrera¹, G.M. Martínez-González², J.L. Navarrete-Bolaños¹, E.O. Castrejón-González²

¹ Departamento de Ingeniería Bioquímica. *Corresponding author. E-mail: hugo.jimenez@itcelaya.edu.mx. Tel. +52 (461) 611-75-75.

³ Departamento de Ciencias Básicas. Instituto Tecnológico de Celaya. Av. Tecnológico y Antonio García Cubas sn. Celaya, Guanajuato. C.P.38010. México.

Received May 13, 2013.
Accepted January 21, 2014.

Abstract

Computational code IMPLI-C3 is a low-RAM consumption program designed to solve three-dimensional parabolic partial differential nonlinear equations. The spatial coordinates are discretized using orthogonal collocation with Legendre polynomials while time was discretized via backward finite differences, generating an implicit method that originates a set of algebraic equations, which are solved by nonlinear relaxation for each step of time integration. Nonlinear relaxation is an iterative method that only uses the Jacobian diagonal and voids the RAM storage of the entire Jacobian matrix. This allows the simulation of physical systems that require greater number of nodes that otherwise would use too much RAM when trying to solve by Newton-Raphson. The code was successfully evaluated using several problems related to natural convection previously reported in literature, observing that nonlinear relaxation only requires 0.3%-1.5% of the memory required by Newton-Raphson for the same problems. Furthermore, one can be conclude that, in problems with many nodes, the use of multivariate Newton-Raphson is unfeasible due to high consumption of RAM that can even cause it to overflow.

Keywords: nonlinear relaxation, natural convection, orthogonal collocation, parabolic partial differential equations.

IMPLI-C3 es un código computacional que utiliza bajos recursos de memoria RAM, diseñado para resolver ecuaciones diferenciales parciales parabólicas no lineales en tres dimensiones. Los ejes coordenados en el espacio se discretizaron usando colocación ortogonal con polinomios de Legendre mientras que el tiempo se discretizó mediante diferencias finitas hacia atrás, generando un esquema implícito que origina un conjunto de ecuaciones algebraicas, las cuales se resolvieron mediante relajación no lineal para cada etapa de integración en el tiempo. La relajación no lineal, es un método iterativo que emplea solamente la diagonal del Jacobiano para evitar que se almacene toda la matriz Jacobiana en la memoria RAM de la computadora. Lo anterior permite la simulación de sistemas físicos que requieren mayor cantidad de nodos que, de otra manera emplearían demasiada RAM al intentar resolverlos mediante Newton-Raphson. El código se evaluó satisfactoriamente usando varios problemas relacionados con el fenómeno de convección natural previamente reportados en la literatura, observando que el método de relajación no lineal solamente utiliza entre 0.3% a 1.5% de la memoria con respecto al método de Newton-Raphson. Además se pudo corroborar que, en problemas con muchos nodos, el uso de Newton-Raphson multivariable no es factible debido al consumo elevado de memoria RAM que, inclusive puede provocar su desbordamiento.

Palabras clave: relajación no lineal, convección natural, colocación ortogonal, ecuaciones diferenciales parciales parabólicas.

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Acknowledgments

We gratefully acknowledge the financial support of CONACYT, Mexico via grants Basic Science SEP-2004-CO1-46230 and CB-2011/167095, and CONCYTEG (Guanajuato State Government) via grants 07-09-k662-050, 07-09-k119-107, and 09-09k662-065.

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