The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

Thin liquid or gas films are everywhere in nature, from foams to submillimetric bubbles at a free surface, and their rupture leaves a collection of small drops and bubbles. However, the mechanisms at play responsible for the bursting of these films is still in debate. The present study thus aims at understanding the drainage dynamics of the thin air film squeezed by gravity between a millimetric droplet and a smooth solid or a liquid thin film. Solving coupled lubrication equations and analyzing the dominant terms in the solid- and liquid-film cases, we explain why the drainage is much faster in the liquid-film case, leading often to a shorter coalescence time, as observed in recent experiments.

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“…1 and 2. Doing so, we get rid of the stiffness of these equations, coming from the high-order spatial derivatives (40,41).…”

Section: Methodsmentioning

confidence: 99%

Dimple drainage before the coalescence of a droplet deposited on a smooth substrate

Duchemin

Josserand

2020

Proc. Natl. Acad. Sci. U.S.A.

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“…As a prototype, let us consider the following 1D axisymmetric mean curvature motion problem [9]: The presence of the u xx guarantees that (2.1) is stiff, suggesting that an implicit time stepping scheme would prove more efficient. However, having to additionally handle the factor of (1 + u 2 x ) −1 complicates the linear algebra.…”

Section: Prototype 1d Problemmentioning

confidence: 99%

“…In each of the references mentioned in the previous paragraph, the authors have succeeded in implementing only a first order time stepping method. Recently in [9], Duchemin and Eggers consolidated the approach and produced a second order linearly stabilized scheme they refer to as the explicit-implicit-null (EIN) method. Their method attains second order accuracy by extrapolating the first order results.…”

Section: Introductionmentioning

confidence: 99%

Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs

Chow

Ruuth

2021

J Sci Comput

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In many applications, the governing PDE to be solved numerically will contain a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this thesis, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods.

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“…The similar idea has also been adopted, for example, by Smereka [22] in the context of flow by mean curvature and surface diffusion, by Jin and Filbet [17] in the context of the Boltzmann equation of rarefied gas dynamics when the Knudsen number is very small, in the context of hyperbolic systems with diffusive relaxation [4], and for the solution of PDEs on surfaces [21]. In a recent study, Duchemin and Eggers [15] proposed to call this method as explicit-implicit-null (EIN) method.…”

Section: Introductionmentioning

confidence: 99%

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

et al. 2019

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In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut = (a(U )Ux)x. The basic idea is to add and subtract two equal terms a0Uxx on the right hand side of the partial differential equation, then to treat the term a0Uxx implicitly and the other terms (a(U )Ux)x − a0Uxx explicitly. We give stability analysis for the method on a simplified model by the aid of energy analysis, which gives a guidance for the choice of a0, i.e, a0 ≥ max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes. The optimal error estimate is also derived for the simplified model, and numerical experiments are given to demonstrate the stability, accuracy and performance of the schemes for nonlinear diffusion equations.

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The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

Cited by 43 publications

References 24 publications

Dimple drainage before the coalescence of a droplet deposited on a smooth substrate

Dimple drainage before the coalescence of a droplet deposited on a smooth substrate

Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

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