Orthogonal cubic spline collocation method for the Cahn–Hilliard equation
Introduction
We consider the one spatial dimensional Cahn–Hilliard equation:with the initial conditionand boundary conditionswhere γ > 0, ϕ(u) = γ2u3 + γ1u2 + γ0u, γ2 > 0, I = (0, 1) and 0 < T < ∞.
Eq. (1.1) arises in a variety of applications such as phase transition in material science. We refer the reader to [5] and the references there in. In this paper, we use second–order splitting procedure combined with orthogonal spline collocation method for Eq. (1.1) and derive optimal error estimates. Since this method is much superior to B-splines in terms of stability, efficiency and conditioning of the resulting matrix. Compared to finite element method (FEM) the calculation of the coefficients of the mass and stiffness matrices determining the approximate solution is very fast since no integrals need to be evaluated or approximated. We discuss numerical experiments using monomial basis functions and RADAU 5 time integrator.
Earlier, mixed methods in combination with orthogonal spline collocation methods used to fourth order evolution equations by Li et al. [9], Manickam et al. [12], [13], Danumjaya and Pani [3]. In the context of Cahn–Hilliard equation, Elliott et al. [6] discussed a second order splitting combined with lumped mass finite element method and derived optimal error estimates.
We split Eq. (1.1) by setting v = γuxx − ϕ(u) then we obtain the following system:with initial conditionand the boundary conditionsWe use orthogonal cubic spline collocation method for the system (1.4), (1.5), (1.6), (1.7) in the spatial direction to compute the approximate solutions using monomial basis functions. Let denote a partition of withandAssume that the partition is quasi-uniform, i.e., there exists a finite positive constant σ such thatWe define a finite dimensional subspace aswhere P3 denotes the set of all cubic polynomials. Let denote the roots of the Legendre polynomial of degree 2 i.e., . These are the nodes of the 2-point Gaussian quadrature rule on the interval I with corresponding weights wk = 1/2, k = 1, 2. Now, we define the collocation pointBelow, we define discrete innerproduct and its induced norm. For any , the discrete innerproduct is defined aswhereand its induced discrete norm by Lemma 1.1 For ,where (respectively, ), is the third derivative of wj (respectively, zj) which is constant on each subinterval Ij. Note that when z = w with , we have
For a proof of Lemma 1.1, we refer to Douglas and Dupont [4].
The outline of the paper is as follows. In Section 1, we introduce some notations and preliminaries. Section 2 deals with continuous-time orthogonal cubic spline collocation method for the solution of (1.4), (1.5), (1.6), (1.7). We establish optimal error estimates. Finally, Section 3 is devoted to numerical experiments. Here, we show that both theoretical order of convergence and numerically computed order of convergence are same.
Throughout this paper, C denotes a generic positive constant which is independent of the discretization parameter h which may have different values at different places.
Section snippets
Continuous-time orthogonal cubic spline collocation method
The continuous-time orthogonal cubic spline collocation approximation to the solution {u, v} of (1.4), (1.5) is a pair of differentiable maps such that for j = 1, 2, … , N and k = 1, 2with appropriate initial approximation U(0) = U(x, 0), which we shall define later. The corresponding discrete Galerkin formulation is written asThe consistent initial condition V(0)
Numerical experiments
In this section, we use orthogonal cubic spline collocation method to approximate the problem (1.4), (1.5), (1.6), (1.7) and we discuss some numerical results. The approximate solution is defined as a pair of differentiable maps satisfying (2.1), (2.2). As in Robinson and Fairweather [14], we use the monomial basis functions to represent U and V, respectively, asandwhere,
Acknowledgements
The first author sincerely thank AR & DB, Ministry of Defence, Govt. of India for financial support through the project “Nonlinear Hyperbolic Waves in Multi-Dimensions with special reference to Sonic Booms” (Ref No: DARO/08/1031199/M/I).
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