Implicit time stepping typically requires solution of one or several linear systems with a matrix I - τJ per time step where J is the Jacobian matrix. If solution of these systems is expensive, replacing I - τJ with its approximate matrix factorization (AMF) (I - τR)(I - τV), R + V =J, often leads to a good compromise between stability and accuracy of the time integration on the one hand and its efficiency on the other hand. For example, in air pollution modeling, AMF has been successfully used in the framework of Rosenbrock schemes. The standard AMF gives an approximation to I - τJ with the error τ2RV, which can be significant in norm. In this paper we propose a new AMF. In assumption that -V is an M-matrix, the error of the new AMF can be shown to have an upper bound τ||R||, while still being asymptotically O(τ2). This new AMF, called AMF+, is equal in costs to standard AMF and, as both analysis and numerical experiments reveal, provides a better accuracy. We also report on our experience with another, cheaper AMF and with AMF-preconditioned GMRES. © 2003 Elsevier B.V. All rights reserved.
Botchev, M. A., & Verwer, J. G. (2003). A new approximate matrix factorization for implicit time integration in air pollution modeling. Journal of Computational and Applied Mathematics, 157(2), 309–327. https://doi.org/10.1016/S0377-0427(03)00414-X