Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Kanney, Joseph; Miller, Cass T.; Kelley, C. T.

doi:10.1016/s0309-1708(02)00162-8

Cited by 98 publications

(67 citation statements)

References 70 publications

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“…The idea is to repeat the splitting steps using the improved computed solutions. Instead of using direct methods, we have to solve a fixed-point iteration and obtain higher-order results [1], [3].…”

Section: Iterative Operator-splitting Methods For Wave Equationsmentioning

confidence: 99%

Seismic sources and waves using iterative operator splitting methods

Geiser

2007

Proc Appl Math and Mech

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Section: Iterative Operator-splitting Methods For Wave Equationsmentioning

confidence: 99%

Seismic sources and waves using iterative operator splitting methods

Geiser

2007

Proc Appl Math and Mech

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“…The linearization with the fixed-point scheme can be used for smooth or weak nonlinear operators, otherwise we lose the convergence behavior, while we did not converge to the local fixed point [26].…”

mentioning

confidence: 88%

“…Newton's method is used to solve the nonlinear parts of the iterative operator-splitting method (see the linearization techniques in [26,28]). We apply the iterative operator-splitting method and obtain:…”

Section: 2mentioning

confidence: 99%

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Operator-splitting methods respecting eigenvalue problems for shallow shelf equations with basal drag

Geiser¹,

Calov²

2012

Coupled Systems Mechanics

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Abstract.We discuss different numerical methods for solving the shallow shelf equations with basal drag. The coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than operator of the basal shear stress. Therefore, we propose a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a stronger iteration on the operator of the membrane stress. We show that this splitting improves the computational performance of the numerical method, although the choice of the (standard) method to solve for all operators in one step speeds up the scheme too. (Based on the delicate and coupled equation we propose a new decomposition method to decouple into simpler solvable sub-equations. After a number of approximations we consider the error of the method and proposed a choice of the operators.)Keywords: partial differential equations, operator-splitting methods, iterative methods, eigenvalue approach.

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“…This is mainly due to the errors introduced by the decoupled equations. Several schemes have been proposed to improve the order of convergence; see for example [12] [13] [36]. All these methods introduce an iterative-type operator splitting that couples the different physical processes and therefore reduces the splitting errors.…”

Section: D Gaussian Hillmentioning

confidence: 99%

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

Oliveira

Afonso

Zlotnik

2016

Computer Methods in Applied Mechanics and Engineering

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This paper presents a particle-based Lagrangian-Eulerian algorithm for the solution of the unsteady advectiondiffusion-reaction heat transfer equation with phase change. The algorithm combines a Lagrangian formulation for the advection + reaction problem with the Eulerian-based heat source method for the diffusion + phase change problem. The coupling between the Lagrangian and Eulerian subproblems is achieved with a phase change detector scheme based on a local latent heat balance and a consistent/conservative interpolation technique between Lagrangian particles and the Eulerian grid. This technique makes use of an auxiliary (finer) Eulerian grid that provides a simple and efficient method of tracking internal heterogeneities (e.g. phase boundaries), allows the use of higher order integration quadratures, and facilitates the implementation of multiscale techniques. The performance of the proposed algorithm is compared against one-and two-dimensional benchmark problems, i.e. pure rigid-body advection, isothermal and non-isothermal phase change, two-phase advective heat transfer and chemical reactions coupled with diffusion and advection. The numerical results confirm that the proposed solution method is accurate, oscillation-free and useful for and applicable to a wide range of fully coupled problems in science and engineering.

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Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Cited by 98 publications

References 70 publications

Seismic sources and waves using iterative operator splitting methods

Seismic sources and waves using iterative operator splitting methods

Operator-splitting methods respecting eigenvalue problems for shallow shelf equations with basal drag

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

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