An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of re
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An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type C. Clavero1
· J. C. Jorge2
Received: 21 February 2019 / Accepted: 7 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this work we are interested in the numerical approximation of the solutions to 1D semilinear parabolic singularly perturbed systems of reaction-diffusion type, in the general case where the diffusion parameters for each equation can have different orders of magnitude. The numerical method combines the classical central finite differences scheme to discretize in space and a linearized fractional implicit Euler method together with a splitting by components technique to integrate in time. In this way, only tridiagonal linear systems must be solved to compute the numerical solution; consequently, the computational cost of the algorithm is considerably less than that of classical schemes. If the spatial discretization is defined on appropriate nonuniform meshes, the method is uniformly convergent of first order in time and almost second order in space. Numerical results for some test problems are presented which corroborate in practice the uniform convergence and the efficiency of the algorithm. Keywords Semilinear parabolic systems · Linearly implicit methods · Splitting by components · Nonuniform meshes · Uniform convergence Mathematics Subject Classification (2010) 65N06 · 65N12 · 65M06
C. Clavero
[email protected] J. C. Jorge [email protected] 1
Department of Applied Mathematics and IUMA, University of Zaragoza, Zaragoza, Spain
2
Department of Computational and Mathematical Engineering and ISC, Public University of Navarra, Pamplona, Spain
Numerical Algorithms
1 Introduction In this paper we consider the numerical resolution of parabolic initial and boundary value problems posed as follows: Find u(x, t) : [0, 1] × [0, T ] → Rn solution of
Lε (u) ≡ ∂u ∂t (x, t) + Lx,ε u(x, t)+A(x, t, u) = 0, (x, t) ∈ Q ≡ Ω × (0, T ], u(0, t) = g0 (t), u(1, t) = g1 (t), ∀ t ∈ [0, T ], u(x, 0) = ϕ(x), ∀ x ∈ Ω, (1)
where Ω = (0, 1) and the spatial differential operator Lx,ε is defined by Lx,ε ≡ −Dε
∂2 , with Dε = diag (ε1 , ε2 , . . . , εn ). ∂x 2
(2)
Let us denote by u = (u1 , u2 , . . . , un )T the exact solution of the continuous problem, g0 = (g10 , g20 , . . . , gn0 )T , g1 = (g11 , g21 , . . . , gn1 )T the boundary data, A(x, t, u) = (a1 (x, t, u), a2 (x, t, u), . . . , an (x, t, u))T the reaction term and ε = (ε1 , ε2 , . . . , εn )T . We assume that the εk , k = 1, . . . , n, can be very small, they can have different orders of magnitude and satisfy 0 < ε1 ≤ ε2 ≤ . . . ≤ εn ≤ 1. As well, we assume that the terms ak (x, t, u), k = 1, . . . , n, are composed of sufficiently smooth functions and that sufficient compatibility conditions among the data of the problem hold, in order to u ∈ C 4,2 ([0, 1] × [0, T ]), (see [15, 17] for a detailed discussion). In many previous works (see [4, 6, 8, 10–12, 18, 19] and re
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