TY - JOUR

T1 - Quasi-Newton methods for the acceleration of multi-physics codes

AU - Haelterman, Rob

AU - Bogaers, Alfred

AU - Degroote, Joris

AU - Boutet, Nicolas

N1 - Funding Information:
R. Haelterman is associate professor at the Royal Military Academy, Dept. Mathematics, Renaissancelaan 30, B-1000 Brussels, Belgium. email: [email protected] A. Bogaers is senior researcher at the Council for Scientific and Industrial Research, Advanced Mathematical Modelling, Modelling and Digital Sciences, Meiring Naudé Road; Brummeria, Pretoria, South Africa and a visiting lecturer at the School of Computer Science and Applied Mathematics, University of Witwatersrand, Johannesburg, South Africa. email: [email protected] J. Degroote is associate professor at Ghent University, Dept. Flow, Heat and Combustion Mechanics, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium. email: [email protected] N. Boutet is PhD student at Ghent University, Dept. Flow, Heat and Combustion Mechanics, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium and Royal Military Academy, Dept. Mathematics, Renaissancelaan 30, B-1000 Brussels, Belgium. email: [email protected]

PY - 2017/8/23

Y1 - 2017/8/23

N2 - Often in nature different physical systems interact which translates to coupled mathematical models. Even if powerful solvers often already exist for problems in a single physical domain (e.g. structural or fluid problems), the development of similar tools for multi-physics problems is still ongoing. When the interaction (or coupling) between the two systems is strong, many methods still fail or are computationally very expensive. Approaches for solving these multi-physics problems can be broadly put in two categories: monolithic or partitioned. While we are not claiming that the partitioned approach is panacea for all coupled problems, here we will only focus our attention on studying methods to solve (strongly) coupled problems with a partitioned approach in which each of the physical problems is solved with a specialized code that we consider to be a black box solver and of which the Jacobian is unknown. We also assume that calling these black boxes is the most expensive part of any algorithm, so that performance is judged by the number of times these are called. Running these black boxes one after another, until convergence is reached, is a standard solution technique and can be considered as a non-linear Gauss-Seidel iteration. It is easy to implement but comes at the cost of slow or even conditional convergence. A recent interpretation of this approach as a rootfinding problem has opened the door to acceleration techniques based on quasi-Newton methods. These quasi-Newton methods can easily be "strapped onto" the original iteration loop without the need to modify the underlying code and with little extra computational cost. In this paper, we analyze the performance of ten acceleration techniques that can be applied to accelerate the convergence of a non-linear Gauss-Seidel iteration, on different multi-physics problems.

AB - Often in nature different physical systems interact which translates to coupled mathematical models. Even if powerful solvers often already exist for problems in a single physical domain (e.g. structural or fluid problems), the development of similar tools for multi-physics problems is still ongoing. When the interaction (or coupling) between the two systems is strong, many methods still fail or are computationally very expensive. Approaches for solving these multi-physics problems can be broadly put in two categories: monolithic or partitioned. While we are not claiming that the partitioned approach is panacea for all coupled problems, here we will only focus our attention on studying methods to solve (strongly) coupled problems with a partitioned approach in which each of the physical problems is solved with a specialized code that we consider to be a black box solver and of which the Jacobian is unknown. We also assume that calling these black boxes is the most expensive part of any algorithm, so that performance is judged by the number of times these are called. Running these black boxes one after another, until convergence is reached, is a standard solution technique and can be considered as a non-linear Gauss-Seidel iteration. It is easy to implement but comes at the cost of slow or even conditional convergence. A recent interpretation of this approach as a rootfinding problem has opened the door to acceleration techniques based on quasi-Newton methods. These quasi-Newton methods can easily be "strapped onto" the original iteration loop without the need to modify the underlying code and with little extra computational cost. In this paper, we analyze the performance of ten acceleration techniques that can be applied to accelerate the convergence of a non-linear Gauss-Seidel iteration, on different multi-physics problems.

KW - Iterative methods

KW - Partitioned methods

KW - Quasi- Newton

UR - http://www.scopus.com/inward/record.url?scp=85027962900&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85027962900

SN - 1992-9978

VL - 47

SP - 352

EP - 360

JO - IAENG International Journal of Applied Mathematics

JF - IAENG International Journal of Applied Mathematics

IS - 3

ER -