A note on B-stability of splitting methods

Araújo, A.

doi:10.1007/bf02663033

Cited by 1 publication

(3 citation statements)

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“…The difference between the two solutions is denoted by

d (MathClass-bin\cdot) MathClass-rel= (MathClass-bin\cdot) MathClass-bin- \overset{(MathClass-bin\cdot)}{true}

. Then, contractivity is defined as

\frac{d}{dt} {(∥∥, d bold-italicχ)}_{scriptH} ⩽ 0 MathClass-punc.

…”

Section: A Sequential Scheme In Numerical Simulationmentioning

confidence: 99%

“…In this section, for completeness, we will find that the fixed‐stress split modified for the multiple porosity model is still B‐stable. As an a priori estimate in numerical stability, we employ B‐stability , defined as

{(∥∥, d χ^{n MathClass-bin+ 1})}_{scriptH} ⩽ {(∥∥, d χ^{n})}_{scriptH} MathClass-punc,

where the corresponding mathematical statements satisfy contractivity. Following the definition of B‐stability, we first investigate contractivity of the fixed‐stress operator splitting modified for the multiple porosity model.…”

Section: A Sequential Scheme In Numerical Simulationmentioning

confidence: 99%

“…In this section, for completeness, we will find that the fixed-stress split modified for the multiple porosity model is still B-stable. As an a priori estimate in numerical stability, we employ B-stability [1,[48][49][50][51][52], defined as…”

Section: An a Priori Stability Estimate Of The Sequential Schemementioning

confidence: 99%

See 2 more Smart Citations

Formulation and sequential numerical algorithms of coupled fluid/heat flow and geomechanics for multiple porosity materials

Kim

Sonnenthal

Rutqvist

2012

Numerical Meth Engineering

100

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SUMMARY We extend constitutive relations of coupled flow and geomechanics for the isothermal elastic double porosity model by Berryman [Journal of Engineering Mechanics ASCE 2002; 128(8):840–847] in the previous study to those for the nonisothermal elastic/elastoplastic multiple porosity model, finding coupling coefficients and constraints of the multiple porosity model and determining the upscaled elastic/elastoplastic moduli as well as relations between the local strains of all materials within a gridblock and the global strain of the gridblock. Furthermore, the coupling equations and relations between local and global variables provide well‐posed problems, implying that they honor the dissipative mechanism of coupled flow and geomechanics. For numerical implementation, we modify the fixed‐stress sequential method for the multiple porosity model. From the a priori stability estimate, the sequential method provides numerical stability when an implicit time‐stepping algorithm is used. This sequential scheme can easily be implemented by using a modified porosity function and its porosity correction. In numerical examples, we observe clear differences among the single, double, and multiple porosity systems, and the multiple porosity model can reflect the substantial heterogeneity that exists within a gridblock. We also identify considerably complicated physics in coupled flow and geomechanics of the multiple porosity systems, which cannot accurately be detected in the uncoupled flow simulation. Copyright © 2012 John Wiley & Sons, Ltd.

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“…The difference between the two solutions is denoted by