ON REACTION-DIFFUSION SYSTEM APPROXIMATIONS ... - CiteSeerX

Proceedings of the Czech–Japanese Seminar in Applied Mathematics 2005 Kuju Training Center, Oita, Japan, September 15-18, 2005 pp. 117–125

ON REACTION-DIFFUSION SYSTEM APPROXIMATIONS TO THE CLASSICAL STEFAN PROBLEMS HIDEKI MURAKAWA Abstract. We consider reaction-diffusion system approximations to the classical two-phase Stefan problem. One of such approximations has been proposed by Hilhorst et al. from ecological points of view. They have given convergence results. We propose a new reaction-diffusion system approximation based on regularization of the enthalpy-temperature constitutive relation. We investigate rates of convergence for each reaction-diffusion system in order to compare two systems. Some numerical experiments are carried out to evaluate the actual rates of convergence. Key words. Stefan problem, Reaction-diffusion systems, convergence rates, regularization AMS subject classifications. 35K57, 35R35, 41A25, 80A22

1. Introduction. Heat transfer problems involving phase change arise in a number of important physical and industrial contexts. A typical model of such problems is the so-called classical two-phase Stefan problem, which describes melting or freezing of a substance in a rather simplified way, by accounting for heat diffusion in each phase and exchange of latent heat at the phase interface. Let Ω ⊂ RN (N ∈ N) be a bounded domain with smooth boundary ∂Ω. The domain Ω is divided into liquid and solid phases by unknown interface Γ(t) at time t ∈ (0, T ), where T is a positive constant. These phases are denoted by Ωu (t) and Ωv (t) respectively. Heat flow occurs in both the liquid and solid phases:  ∂u   = d1 ∆u in Ωu := ∪0 η, ξr if 0 ≤ r ≤ η, β ξ (r) =  d2 r if r < 0.

The weak solution of (2.1) is H¨ older continuous in Q (see [10]), although the weak solution of (SP) is generally discontinuous. In this sense, we can interpret (RD)ξ as the combination of a reaction-diffusion system approximation and a regularization procedure. We make the following assumption on the initial data  ε,ξ ε,ξ u0 , v0 , w0ε,ξ ∈ C(Ω), (2.2) 0 ≤ uε,ξ 0 ≤ v0ε,ξ ≤ M, 0 ≤ w0ε,ξ ≤ η in Ω 0 ≤ M,

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H. Murakawa

for some positive constant M independent of ε and ξ. Then, the existence and uniqueness of the solution of (RD)ξ follows from Lunardi [11, Proposition 7.3.2]. Now, we propose our results on convergence rates for the reaction-diffusion systems (RD)ξ , ξ ≥ 0. Theorem 2.1. Let z be the unique weak solution of (SP) with an initial datum z0 ∈ L∞ (Ω) and (uε , v ε , wε ) be the unique solution of (RD)0 with initial data satisfying assumption (1.6). Let us set z ε = uε − v ε + wε . Then, there exists a positive constant C independent of ε such that

Z t

ε ε ε

2 2 ku − u kL (Q) + kv − v kL (Q) + (θ − θ ) 0

ε

L∞ (0,T ;H 1 (Ω))

+kz − z kL∞(0,T ;(H 1 (Ω))∗ ) ≤ C(ε

1/2

+ σ(ε))1/2 ,

where u := φ1 (z), v := φ2 (z), θ := β(z) := d(φ(z)), θ ε := d1 uε − d2 v ε , σ(ε) := kz0 − z0ε k2L2 (Ω) and z0ε = uε0 − v0ε + w0ε . The rate of convergence is O(ε1/4 ) if σ is of order ε1/2 . Theorem 2.2. Let z be the unique weak solution of (SP) with an initial datum z0 ∈ L∞ (Ω) and (uε,ξ , v ε,ξ , wε,ξ ) be the unique solution of (RD)ξ with initial data satisfying assumption (2.2). Let us set z ε,ξ = uε,ξ − v ε,ξ + wε,ξ . Then, there exists a positive constant C independent of ε and ξ such that ku−uε,ξ kL2 (Q) + kv − v ε,ξ kL2 (Q) + ξ 1/2 kz − z ε,ξ kL2 (Q)

Z t

ε,ξ

+ (θ − θ ) + kz − z ε,ξ kL∞ (0,T ;(H 1 (Ω))∗ ) ∞ 1 0 L (0,T ;H (Ω)) ξ ε ≤ C(ξ A (z) + ξ 2 + + σ(ε, ξ))1/2 , ξ

where u = φ1 (z), v = φ2 (z), θ = β(z), θ ε,ξ := d1 uε,ξ − d2 v ε,ξ + ξwε,ξ , σ(ε, ξ) := kz0 − ε,ξ ε,ξ ξ z0ε,ξ k2L2 (Ω) , z0ε,ξ := uε,ξ 0 − v0 + w0 and A (z) := {(x, t) ∈ Q | 0 ≤ β(z(x, t)) ≤ ηξ}. The general rate of convergence is O(ε1/4 ) if we choose ξ and σ are of order ε1/2 . If the non-degeneracy property |Aξ (z)| ≤ Cξ for some positive constant C is valid, which was studied by Nochetto [14], and we choose ξ is of order ε1/3 and σ is of order ε2/3 , then the rate becomes O(ε1/3 ). In addition, the corresponding latent heat wε,ξ and the corresponding enthalpy z ε,ξ converge strongly in L2 (Q). The rate of convergence is O(ε1/6 ). Thus, we get better results for the reaction-diffusion system (RD)ξ than for (RD)0 . 3. Numerical experiments. The aim of our numerical experiments are to evaluate the actual rates of convergence of the temperature and the enthalpy, as well as the interface. We treat a one-dimensional problem with the domain Ω = (0, 1). An analytical solution is constructed to the classical formulation of the Stefan problem with a prescribed interface Γ(t) which separates the liquid (0, Γ(t)) and the solid (Γ(t), 1) regions. Let T = 1 and λ = 1. Forcing functions are chosen such that exact temperatures of liquid u and solid −v are given by 1 (Γ(t) − x) + d2 (Γ(t) − x)2 , d1 1 v(x, t) = − (Γ(t) − x) + d1 (Γ(t) − x)2 , 2d2 u(x, t) =

(x, t) ∈ Ωu := ∪0