A Jacobian-free Newton-Krylov method for thermalhydraulics simulations

• Published in: International journal for numerical methods in fluids Vol. 77; no. 10; pp. 590 - 615
• Main Authors: Ashrafizadeh, A., Devaud, C.B., Aydemir, N.U.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 10-04-2015; Wiley Subscription Services, Inc
• Subjects: Accuracy; Algorithms; Computation; Discretization; implicit; Jacobi matrix method; Jacobian matrix; Jacobian-free Newton-Krylov; Mathematical analysis; Mathematical models; Solvers; thermalhydraulics; two-phase flows
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.3999

SummaryThe current paper is focused on investigating a Jacobian‐free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two‐phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two‐fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well‐known benchmarks are considered, the water faucet and the oscillating manometer.An isentropic two‐fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed.For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.
The fully implicit Jacobian‐Free Newton‐Krylov (JFNK) method is applied to the four equation two‐fluid model. A detailed analysis is performed. High Courant‐Friedrichs‐ Lewy (CFL) numbers can be used to obtain stable numerical results and the good accuracy of the transient and steady‐state predictions is maintained with significant decrease in computational time.