Numerical Approximation of the Phase-Field Transition System With Non-Homogeneous Cauchy-Neumann Boundary Conditions in Both Unknown Functions via Fractional Steps Method

Ovono, Armel Andami

doi:10.11948/2013028

Cited by 7 publications

(10 citation statements)

References 9 publications

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“…where φ ε − stands for the left-hand limit of φ ε . Detailed discussions with respect to the advantage of ( 23)-(25) can be found in the works [3,4,15,17,18].…”

Section: Approximating Scheme-convergencementioning

confidence: 99%

“…Next, we are interested in the convergence of the sequence (θ ε , α ε ), (φ ε , ξ ε ) of solutions to ( 23) and ( 24) to (θ, α), (φ, ξ) -the solution of problems ( 6) and ( 7) (see [3,17,18,20] for more details).…”

Section: Approximating Scheme-convergencementioning

confidence: 99%

“…• K 1 s, y, θ(s, y) and K 2 s, y, φ(s, y) are the mobility functions (attached to the solution θ(s, y), φ(s, y), (s, y) ∈ Q, of (1) 1 and (1) 2 , respectively; see [2] for more details); • f 1 (t, x) ∈ L p (Q) and f 2 (t, x) ∈ L q (Q) are given functions (see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16] for more details).…”

Section: Introductionmentioning

confidence: 99%

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On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

Fetecău,

Moroşanu,

Pavăl

2024

Axioms

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The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, α, ξ, θ0, α0, φ0, and ξ0, we prove the well-posedness of a solution θ,α,φ,ξ, which is θ(t,x),α(t,x)∈Wp1,2(Q)×Wp1,2(Σ), φ(t,x),ξ(t,x)∈Wν1,2(Q)×Wp1,2(Σ), ν=min{q,μ}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.

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“…where φ ε − stands for the left-hand limit of φ ε . Detailed discussions with respect to the advantage of ( 23)-(25) can be found in the works [3,4,15,17,18].…”

Section: Approximating Scheme-convergencementioning

confidence: 99%

Section: Approximating Scheme-convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

Fetecău,

Moroşanu,

Pavăl

2024

Axioms

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“…where ϕ ε − stands for the left-hand limit of ϕ ε . For a detailed discussion regarding the importance of the above numerical scheme, we direct the reader to the works [1,[10][11][12]14].…”

Section: Approximating Scheme Of Fractional Steps Type: the Phase-fie...mentioning

confidence: 99%

Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions

Fetecau,

Moroşanu,

Stoicescu

2023

Axioms

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Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class Wp1,2(Q)×Wν1,2(Q). One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, alg-frac_sec-ord-varphi_PHT, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper.

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“…Problem ( 1) is a particular instance of the Allen-Cahn equation [18], which was introduced to describe the motion of anti-phase boundaries in crystalline solids, and it has been widely applied to many complex moving interface problems, e.g., the mixture of two incompressible fluids, the nucleation of solids and vesicle membranes. Also, the nonlinear problem (1) occurs in the phase-field transition system where the phase function V (t, x) describes the transition between the solid and liquid phases in the solidification process of a material occupying a region Ω (see [1,2,4,5,10,11,12,13,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]). For more general assumptions and with various types of boundary conditions, equation (1) has been numerically investigated in, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,17,18,19,20,21,…”

mentioning

confidence: 99%

A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions

Tănase

2024

DCDS-S

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In this present paper, we consider a nonlinear reaction-diffusion problem (1), endowed with a cubic nonlinear reaction term and homogeneous Cauchy-Neumann boundary conditions. We will approach the proposed nonlinear parabolic problem in the spirit of Hadamard's well-posedness conditions (see [26, p. 46]). Practically, we start our study by investigating the solvability of such a problem in the class W 1,2 p (Q), p ≥ 2. The second goal is to develop an iterative splitting scheme, corresponding to the nonlinear reaction-diffusion problem in question. Results about the convergence of the numerical scheme and error estimation are established, too. On the basis of the proposed numerical scheme, we formulate a conceptual algorithm GTanase-alg-frac rd CN-bc, which represents a delicate challenge for our future works, in order to approximate the solution of the nonlinear parabolic problem (1). The benefit of such a method aims at simplifying the process of numerical computations due to its decoupling feature.• t ∈ (0, T ], x = (x 1 , . . . , x n ) ∈ Ω; • p d , p r , p s and p t are positive values, representing the diffusion, reaction, source, and thermal transfer parameters, respectively; • V (s, y), (s, y) ∈ Q, is the unknown function (the order parameter in Q, for example);

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Numerical Approximation of the Phase-Field Transition System With Non-Homogeneous Cauchy-Neumann Boundary Conditions in Both Unknown Functions via Fractional Steps Method

Cited by 7 publications

References 9 publications

On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions

A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions

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