Thermodynamic equilibrium solutions through a modified Newton Raphson method

Abstract

In numerical codes for reactive transport modeling, systems of nonlinear chemical equations are often solved through the Newton Raphson method (NR). NR is an iterative procedure that results in a sequential solution of linear systems. The algorithm is known for its effectiveness in the vicinity of the solution but also for its lack of robustness otherwise. Therefore, inaccurate initial conditions can lead to non‐convergence or excessive numbers of iterations, which significantly increase the computational cost. In this work, we show that inaccurate initial conditions can lead to very ill‐conditioned system matrices, which makes NR inefficient. This efficiency is improved by preconditioning techniques and/or by coupling the NR method with a zero‐order method called the positive continuous fraction method. Numerical experiments that are based on seven different test cases show that the ill‐conditioned linear systems within NR represent a problem and that coupling NR with a method that bypasses the computation of the Jacobian matrix significantly improves the robustness and efficiency of the algorithm. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1246–1262, 2017

Jacobian Free Methods for Coupling Transport with Chemistry in Heterogenous Porous Media

“…As mentioned previously, we take as unknowns not the aqueous concentrations themselves but their logarithms. This has the obvious advantage that the concentrations will always be positive, and also reduces the ill-conditioning of the non-linear system (see [22,49,50]). We denote by ξ p (resp.…”

“…where Y = diag exp(ξ S ), and I is an identity matrix of the appropriate size. These systems can become severely ill-conditioned (see [49]) so that Gaussian elimination may give inaccurate solutions. We monitor the condition number of the Jacobian and if it is larger than 10 15 , so that the solution may have no correct digit, we perform a singular value decomposition and compute a least squares solution.…”

Algorithms for activity correction models for reactive transport.

“…The methodology for computing chemical equilibrium is well established [13][14][15][16][17][18][19] and is always built on a Newton procedure. By incorporating the Nc mass action laws into the Nx conservation equation, one can obtain an Nx by Nx nonlinear system that must be iteratively solved.…”

“…It includes 3 components and 17 chemical species. It is a classical test case, and many difficulties in convergence have been reported while solving it by using Newton or Newton-like algorithms [13,14,21]. A table including the stoichiometric coefficients, equilibrium constants, total concentrations and equilibrium solutions is given in appendix 3.…”

“…Chromium (VI), which is the most toxic and mobile form of chromium, is reduced by iron (II) to yield chromium (III), which has a much lower solubility and is less toxic [23]. This test is reported to be a very difficult one [14,21], so here we use some favorable testing conditions to increase the convergence of the Newton algorithm. A table including the stoichiometric coefficients, equilibrium constants, total concentrations and equilibrium solutions is given in appendix 4.…”