On the stability of a finite difference scheme with two weights for wave equation with nonlocal conditions

We consider the stability of a finite difference scheme with two weight parameters for a hyperbolic equation with nonlocal integral boundary conditions. We obtain stability region in the complex plane by investigating the characteristic equation of a difference scheme using the root criterion.


Introduction
There often arise problems described by equations of mathematical physics with rather complicated nonclassical conditions modeling natural, physical, chemical and other processes.Nonlocal conditions occur in processes related to diffusion processes, for instance, electrolytic refining of non-ferrous metals [4], deformation of metals under high strain rates, the phenomena of Ohmic heating (see [2] and references therein), superconductivity [1], flow of fluids through fissured rocks [7], etc.
In the present paper, we investigate the stability region of the finite difference scheme (FDS) with two parameters (see [6]) for the hyperbolic equation with two integral nonlocal boundary conditions (NBC).By using the root criterion (see [3]) we obtain regions on a complex plane, where FDS is stable.Samarskii, using the energy inequality technique, obtained the stability conditions for the classical hyperbolic problem in work [6].We have generalized the results presented in [5], by using more general scheme.We note, that, unlike the case of FDS with one weight parameter, the eigenvalues of the investigated problem could be complex.

A finite difference scheme
Consider the wave equation where Ω = (0, L), with the classical initial conditions and integral NBC where f (x, t), φ(x), ψ(x), v l (t), and v r (t) are given functions, and γ 0 and γ 1 are given real parameters.We are interested in sufficiently smooth solutions of the nonlocal problem (1)-( 4).This paper is the generalization of the article [5], therefore in both works we use the same notations.We can investigate problem (1)-( 4) in the interval [0, 1] instead of [0, L] using transformation x = Lx ′ .Then new c ′ = c/L.Further we consider c ′ = 1, without losing of generality, for simplicity.Now we state a difference analogue of the differential problem (1)-( 4).We denote We define a FDS approximating the original differential equation (1) (see [6]): The initial conditions are approximated as follows: We rewrite the boundary conditions using the defined in article [5] inner product: In the problem ( 5)-( 9) we approximate functions f , φ, ψ, v l and v r by grid functions F , Φ, Ψ , V l , and V r .In the case σ 1 = σ 2 = σ stability of FDS ( 5)-( 9) is equal to the one, investigated in [5].Equations ( 8)-( 9) is a system of two linear equations for unknowns U 0 and U N .We express these unknowns via inner points U i , i = 1, N − 1, and obtain where . By substituting expressions (11) and (10) into Eq.( 5) for i = 1 and i = N − 1 we rewrite it in the form Liet. matem.rink.Proc.LMS, Ser.A, 55, 2014, 22-27.
where A, B, C, and are (N − 1) × (N − 1) matrices, I is the identity matrix, 0 is a zero matrix.Finally, The spectrum of matrix Λ is fully investigated in paper [5, §3].According to [5, Lemma 1 and Remark 2] under certain conditions (γ < 2) spectrum is real and is in the interval (0, 4/h 2 ]. We represent the three-layer scheme (11) as an equivalent two-layer scheme (e.g., see [5]) using notations According to [5] eigenvalues µ of the matrix S could be found as the roots of the quadratic equation where λ k are the eigenvalues of the matrix Λ.
The aim of the following section is to investigate the spectrum of the weighted FDS independently of boundary conditions.

FDS stability regions
In general, under various boundary conditions, eigenvalues of operator Λ could be complex numbers.A polynomial satisfies the root condition if all the roots of that polynomial are in the closed unit disc of complex plane and roots of magnitude 1 are simple [3].If polynomial p(µ, λ) := a(λ)µ 2 + b(λ)µ + c(λ) satisfies the root condition, then we say that λ is in stability region defined by equation p(µ, λ) = 0. Denoting z := τ λ and substituting it into (16) we have: or expressing z: Substituting µ = e ıϕ , ϕ ∈ (−π, +π], into Eq.( 17) we obtain the formula for the boundary ∂S of the stability region S: One can see that Re z ∂ is even function and Im z ∂ is odd function.So, the stability region is symmetric to the real axis (see Fig. 1).The discriminant of the polynomial P (µ, z) is We have two double root points on the real axis: and corresponding real values of µ: and z ′ (1) = z ′ (µ 1 ) = 0.These two points are the branch points of the multi-valued function µ(z).Point z 0 = 0 is on the boundary ∂S and corresponds to double root µ = µ 0 = 1 (ϕ = 0).So, this point does not belong to the stability region S.