TWO NEW APPROACHES FOR CONSTRUCTION OF THE HIGH ORDER OF ACCURACY DIFFERENCE SCHEMES FOR HYPERBOLIC DIFFERENTIAL EQUATIONS

We consider the abstract Cauchy problem for differential equation of the hyperbolic type v′′(t) +Av(t) = f (t) (0 ≤ t ≤ T), v(0) = v0, v′(0) = v′ 0 in an arbitrary Hilbert space H with the selfadjoint positive definite operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.


The Cauchy problem
We consider the abstract Cauchy problem for hyperbolic equations in a Hilbert space H with the selfadjoint positive definite operator A.
A function v(t) is called a solution of the problem (1.1) if the following conditions are satisfied.
(i) v(t) is twice continuously differentiable on the segment [0, T].The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.
(ii) The element v(t) belongs to D(A) for all t ∈ [0,T], and the function Av(t) is continuous on the segment [0,T].
(iii) v(t) satisfies the equations and the initial conditions (1.1).
It is known (see, e.g., [7,9]) that various initial boundary value problems for the hyperbolic equations can be reduced to the problem (1.1).A study of discretization, over time only, of the initial value problem also permits one to include general difference schemes in applications, if the differential operator in space variables, A, is replaced by the difference operators A h that act in the Hilbert spaces and are uniformly positive definite and selfadjoint in h for 0 < h ≤ h 0 .
In a large cycle of works on high order of accuracy difference schemes for hyperbolic partial differential equations (see, e.g., [1,10] and the references given therein), stability was established under the assumption that the magnitudes of the grid steps τ and h with respect to the time and space variables are connected.In abstract terms this means, in particular, that the condition τ A h → 0 when τ → 0 is satisfied.
Of great interest is the study of absolute stable difference schemes of a high order of accuracy for hyperbolic partial differential equations, in which stability was established without any assumptions with respect to the grid steps τ and h.In [4,5,12] the corresponding simple difference schemes of the first and second order of accuracy for hyperbolic partial differential equations were studied.
In the present paper the two-step difference schemes of a high order of accuracy generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of the problem (1.1) are presented.The stability estimates for the solutions of these difference schemes are established.In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.
If the function f (t) is not only continuous, but also continuously differentiable on [0,T], v 0 ∈ D(A), and v 0 ∈ D(A 1/2 ), it is easy to show that the formula gives a solution of problem (1.1).Here Actually, obviously (1.1) can be rewritten as the equivalent initial value problem for system of the first-order linear differential equations where B = A 1/2 .Integrating these, we at once obtain (1.7) A. Ashyralyev and P. E. Sobolevskii 185 By the interchange of the order of integration, we obtain v(t) = e −iBt + e iBt − e −iBt 2 v 0 + B −1 e iBt − e −iBt 2i dsv (0) B −1 e iB(t−s) − e −iB(t−s) 2i f (s)ds.
(1.8) Thus, from that by the definitions of B, c(t), and s(t) we have (1.2). Theorem hold, where M does not depend on f (t), t ∈ [0,T], and v 0 , v 0 .
Proof.Using the formula (1.2) and estimates we obtain (1.11) Applying A 1/2 to the formula (1.2) and using the estimates (1.10), in a similar manner with (1.10), we obtain (1.12) Now, we obtain the estimate for Av(t) H . Applying A to the formula (1.2) and using an integration by parts, we can write (1.13) Using the last formula and estimates (1.10), we obtain (1.14) Then from the last estimate, it follows that max 0≤t≤T The estimate for max 0≤t≤T d2 v(t)/dt 2 H follows from the last estimate and the triangle inequality.
Remark 1.2.Theorem 1.1 holds in an arbitrary Banach space E under the following assumptions (see, e.g., [3,8,11] and the references given therein): Now, we consider the application of this abstract Theorem 1.1.First, we consider the mixed problem for wave equation (1.17) The problem (1.17) has a unique smooth solution u(t,x) for the smooth a(x) > 0 (x ∈ [0,L]), ϕ(x), ψ(x) (x ∈ [0,L]), and f (t,x) (t ∈ [0,T], x ∈ [0,L]) functions.This allows us to reduce the mixed problem (1.17) to the initial value problem (1.1) in Hilbert space H = L 2 [0,L] with a selfadjoint positive definite operator A defined by (1.17).We give a number of corollaries of the abstract Theorem 1.1.
Theorem 1.3.For solutions of the mixed problem (1.17), the stability inequalities The proof of this theorem is based on the abstract Theorem 1.1 and the symmetry properties of the space operator generated by the problem (1.17).
We introduce the Hilbert space L 2 (Ω) which is the space of all the integrable functions defined on Ω, equipped with the norm hold, where M does not depend on f (t,x) and ϕ(x), ψ(x).
The proof of this theorem is based on the abstract Theorem 1.1 and the symmetry properties of the space operator generated by the problem (1.19).

The high order of accuracy difference schemes generated by an exact difference scheme
We consider the initial value problem (1.1).On the segment [0,T] we consider a uniform grid with step τ.Using the formula (1.2), we can establish the following two-step difference scheme for the solution of the initial value problem (1.1) 3) The latter will be referred to as the exact two-step difference scheme for the initial value problem (1.1).
Suppose the operators (I − e 2τiB ) and (I + e τiB ) have the bounded inverses (I − e 2τiB ) −1 and (I + e τiB ) −1 .Then this problem is uniquely solvable, and the following formula holds: Actually, (2.3) can be rewritten as the equivalent initial value problem for system of the first-order difference equations: (2.5) A. Ashyralyev and P. E. Sobolevskii 189 From that, the system of recursion formulas follows : Hence (2.7) From that it follows that (2.10) Using this formula and the formula τz 1 = u 1 − e −τiB u 0 , we can obtain the formula (2.4).We investigate the stability of the exact two-step difference scheme (2.3).
Then for the solution of the exact two-step difference scheme (2.3), the following stability inequalities hold: where M does not depend on τ, f k , 1 ≤ k ≤ N − 1, and u 0 , u 1 .
Proof.Using the formula (2.4) and the estimates we obtain for k ≥ 2. It obviously holds also for k = 0,1.The estimate (2.11) is established.Applying τ −1 (I − e −τiB ) to the formula (2.4) and using the estimates (2.14) and A. Ashyralyev and P. E. Sobolevskii 191 in a similar manner with (2.11), we obtain In a similar manner, one can show that The estimate (2.12) is established.Now, we obtain the estimate for (2/τ 2 )(c(τ (2.20) Using the last formula and estimates (2.14) and (2.16), we obtain for k ≥ 2. It obviously holds also for k = 1.So, we have that The estimate for max 1≤k≤N−1 τ −2 (u k+1 − 2u k + u k−1 ) H follows from the last estimate and the triangle inequality.
A. Ashyralyev and P. E. Sobolevskii 193 Note that we have not been able to obtain the following stability inequalities: for the solution of the exact two-step difference scheme (2.3).Nevertheless, the following result holds.
Then for the solution of the exact two-step difference scheme (2.3), the following stability inequalities hold: where M does not depend on τ, f k , 1 ≤ k ≤ N − 1, and u 0 , u 1 .
Proof.Using the formula (2.4) and the estimates (2.14), (2.16) and for k ≥ 3. It obviously holds also for k = 1,2.So, we have that (2.30) In a similar manner, one can show that for k ≥ 2. It obviously holds also for k = 1.So, we have that (2.33) The estimate for max 1≤k≤N−1 τ −2 (u k+1 − 2u k + u k−1 ) H follows from the last estimate and the triangle inequality.
Remark 2.3.Theorem 2.1 actually holds in an arbitrary Banach space E under the following assumptions: where M does not depend on τ.
Remark 2.4.This approach permits us to construct the high order of accuracy two-step uniform difference schemes for differential equations of the hyperbolic type with an arbitrary parameter ε on the highest derivative.In [2,3] the stability estimates of the solutions of the high order of accuracy difference schemes for hyperbolic equations with an arbitrary ε parameter on the highest derivative were obtained.Now, we will consider the applications of the exact difference scheme (2.3).From (2.3) it is clear that for the approximate solutions of the problem (1.1), it is necessary to approximate the expressions We remark that in constructing difference schemes, it is important to know how to construct f j,l k and f j,l 1,1 such that (2.38) A. Ashyralyev and P. E. Sobolevskii 197 In a similar manner, one can obtain that Using the definitions of s(τ) and c(τ) and Pade fractions for the function e −z (see [1]), we can write (2.41) Now, using the formulas (2.36), (2.37), (2.39), and (2.41), f j,l k and f j,l 1,1 can be defined by the following formulas: where (2.43) Now, using the formulas (2.41), (2.42), and (2.37), we obtain the difference schemes ( j + l)th order of accuracy for the approximate solution of the initial value problem (1.1).Note that the difference schemes (2.44) for j = l, j = l − 1, and j = l + 1 include difference schemes of arbitrary high order of approximation.Moreover, the corresponding functions |R j,l+1 (z)| tend to 0 as z → ∞ for j = l − 1,l and |R j,l+1 (z)| = 1 for j = l + 1.Such difference schemes are simplest, in the sense that the degrees of the denominators of the corresponding Pade approximants of the function exp{−z} are minimal for a fixed order of approximation of the difference schemes.
Suppose the operators (I − R 2 j,l (iτB)) and (I + R j,l (iτB)) have the bounded inverses (I − R 2 j,l (iτB)) −1 and (I + R j,l (iτB)) −1 .It is clear that this problem is uniquely solvable, and the following formula holds: We investigate the stability of the exact two-step difference scheme (2.45).
Theorem 2.5.Suppose the operators (I − R 2 j,l (iτB)) and (I + R j,l (iτB)) have the bounded inverses (I − R 2 j,l (iτB)) −1 and (I + R j,l (iτB)) −1 .Suppose that u 0 ∈ D(B 2 ), u 1 − u 0 ∈ D(B).Then for the solution of the two-step difference scheme (2.45) for j = l and j = l − 1, the A. Ashyralyev and P. E. Sobolevskii 199 following stability inequalities hold: where M does not depend on τ, f j,l k , 1 ≤ k ≤ N − 1, and u 0 , u 1 .The proof of Theorem 2.5 follows the scheme of the proof of Theorem 2.1, and relies on the formula (2.45) and on the estimates ) Note that the estimate (2.48) is not satisfied for j = l + 1. Therefore we have not been able to obtain the same stability inequalities for the solution of the exact two-step difference scheme (2.45) for j = l + 1.Nevertheless, the following result holds.
Theorem 2.6.Suppose the operators (I − R 2 j,l (iτB)) and (I + R j,l (iτB)) have the bounded inverses (I − R 2 j,l (iτB)) −1 and (I + R j,l (iτB)) −1 .Suppose that u 0 ∈ D(B 3 ), u 1 − u 0 ∈ D(B 2 ), and Then for the solution of the two-step difference scheme (2.45) for j = l + 1, the following stability inequalities hold: where M does not depend on τ, f 200 High order of accuracy difference schemes for HDE The proof of Theorem 2.6 follows the scheme of the proof of Theorem 2.2, and relies on the formula (2.45), the estimates (2.48), and Remark 2.7.Theorem 2.6 actually holds in an arbitrary Banach space E under the following assumptions: where M does not depend on τ.
Now, the abstract Theorems 2.5 and 2.6 are applied in the investigation of difference schemes of higher order of accuracy with respect to one variable for approximate solutions of the mixed boundary value problem (1.19).The discretization of problem (1.19) is carried out in two steps.In the first step we define the grid sets (2.52) We introduce the Banach space L 2 ( Ω h ) of the grid functions ϕ h (x) = {ϕ(h 1 m 1 ,...,h n m n )} defined on Ω h , equipped with the norm

.53)
To the differential operator A x generated by the problem (1.19), we assign the difference operator A x h by the formula acting in the space of grid functions u h (x), and satisfying the conditions u h (x) = 0 for all x ∈ S h .It is known that A x h is a selfadjoint positive definite operator in L 2 ( Ω h ).With the help of A x h , we arrive at the initial value problem for an infinite system of ordinary differential equations.
A. Ashyralyev and P. E. Sobolevskii 201 In the second step we replace problem (2.56) by the difference schemes (2.44): where (2.57) Theorem 2.8.Let τ and |h| be sufficiently small numbers.Then the solutions of the difference schemes (2.56) for j = l and j = l − 1 satisfy the following stability estimates: 202 High order of accuracy difference schemes for HDE . (2.58) Here M 1 does not depend on τ, h, ϕ h (x), ψ h (x), and The proof of Theorem 2.8 is based on the abstract Theorem 2.2, and the symmetry properties of the difference operator A x h defined by the formula (2.56).Theorem 2.9.Let τ and |h| be sufficiently small numbers.Then the solutions of the difference schemes (2.56) for j = l + 1 satisfy the following stability estimates: . (2.59) Here M 1 does not depend on τ, h, ϕ h (x), ψ h (x), and f h k (x), 0 ≤ k ≤ N − 1.The proof of Theorem 2.9 is based on the abstract Theorem 2.5, and the symmetry properties of the difference operator A x h defined by the formula (2.56).Note that in a similar manner one can construct the difference schemes of a high order of accuracy with respect to one variable for approximate solutions of the boundary value problem (1.17).Abstract Theorems 2.2 and 2.5 permit us to obtain the stability estimates for the solutions of these difference schemes.

The high order of accuracy difference schemes generated by Taylor's decomposition of function on the three points
We consider again the initial value problem (1.1).For the construction of the two-step difference schemes of an arbitrary high order of accuracy for the approximate solutions of the initial value problem (1.1), we consider again a uniform grid space The function v(t) (0 ≤ t ≤ T) has a (2l + 2 j + 2)th continuous derivative and t k−1 ,t k , t k+1 ∈ [0,T] τ .Then we have the following Taylor decomposition on the three points (see [6]): where η m , m = 1,..., j, is the solution of system Suppose further that the function v(t) (0 ≤ t ≤ T) has a (2l + 2 j + 1)th continuous derivative.Then we have the following Taylor decomposition on the two points (see [6]): where Now, we will consider the applications of the Taylor decomposition (3.2) of function v(t) on the three points and Taylor's decomposition (3.4) of function v(t) on the two points to approximate solutions of the initial value problem (1.1).From (3.2) and (3.4) it is clear that for the approximate solution of the problem (1.1), it is necessary to find v ( j) (τ) for any s, 1 ≤ s ≤ 2l, v (2s) (t k ) for any s, 1 ≤ s ≤ l, and v (2s) (t k−1 ), v (2s) (t k+1 ) for any s, 1 ≤ s ≤ j.Using the equation we obtain ) Suppose further that the function v(t) (0 ≤ t ≤ T) has a (2l + 2 j − 2m + 2)th continuous derivative.Then by Theorem 1.3, we have the following Taylor decomposition on the two points: where where [a] denotes the integer part of the number a.Further using the formulas (3.7), (3.8), and (3.9), we can write A. Ashyralyev and P. E. Sobolevskii 205 From the last formula it follows that (3.12) Suppose further that the operator (I + l−[m/2] n=1 a 2n (−A) n τ 2n ) has a bounded inverse.Then (3.13) 206 High order of accuracy difference schemes for HDE Now, using the formulas (3.2), (3.4), (3.7), (3.8), and (3.13), we obtain the difference schemes of a (2l + 2 j)th order of accuracy (3.14) for the approximate solution of the initial value problem (1.1). (3.18) From the formula (3.17) it follows that the investigation of the stability of difference schemes (3.14) relies in an essential manner on a number of properties of the powers of the operator R j,l (±iτA).We were not able to obtain the estimates for powers of the operator R j,l (±iτA) in the general cases of numbers j and l.
Theorem 3.1.Suppose that u 0 ∈ D(A), u 1 − u 0 ∈ D(A 1/2 ).Then for the solution of the two-step difference schemes (3.14) for l = 0 the following stability inequalities hold: where M does not depend on τ, f j,l k , 1 ≤ k ≤ N − 1, and u 0 , u 1 .The proof of Theorem 3.1 follows the scheme of the proof of Theorem 2.1, and relies on the formula (3.17) and on the estimates for the approximate solution of the initial value problem (1.1).In a similar manner one can obtain the stability inequalities for the solution of the two-step difference schemes (3.22).Now, the abstract Theorem 3.1 is applied in the investigation of difference schemes of higher order of accuracy with respect to one variable for approximate solutions of the mixed boundary value problem (1.19).The first step of discretization of problem (1.19) is given above.Suppose that the operator I +  Here M does not depend on τ, h, ϕ h (x), ψ h (x), and f h k (x), 0 ≤ k ≤ N − 1.The proof of Theorem 3.4 is based on the abstract Theorem 3.1, and the symmetry properties of the difference operator A x h is defined by the formula (2.56).Note that in a similar manner one can construct the difference schemes of a high order of accuracy with respect to one variable for approximate solutions of the boundary value problem (1.17).Abstract Theorem 3.1 permits us to obtain the stability estimates for the solutions of these difference schemes.