The spline approach to the calculation of derivatives on the Bakhvalov mesh in the presence of a boundary layer

Abstract

The problem of calculating the derivatives of a function with large gradients in the region of an exponential boundary layer is considered. To do this, it is proposed to construct a cubic spline on the Bakhvalov grid, which thickens in the boundary layer. We study the convergence of the derivatives of the constructed spline to the derivatives of the function given at the grid nodes. Error estimates are obtained taking into account uniformity in a small parameter. The obtained error estimates are confirmed by the results of computational experiments.

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Introduction where

u(x) = p(x) + Φ(x); x 2 [0; 1]; jp(j)(x)j

C1; jΦ(j)(x)j

C"j1 e− x="; 0 j 4; ( 1 ) ( 2 ) Copyright ⃝c by the paper's authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). where the functions p(x) and Φ(x) are not explicitly defined, > 0; " 2 (0; 1], constant C1 is independent of ". According to ( 2 ), the regular component p(x) has bounded derivatives to the fourth order, and the boundary-layer component Φ(x) has derivatives that are not uniformly bounded with the respect to the parameter ". According to [1], the representation ( 1 ) with constraints ( 2 ) is valid for the solution of the boundary value problem for a second-order differential equation with a small parameter " with the highest derivative.

Let us show the relevance of developing difference formulas for calculating derivatives with respect to function values at mesh nodes if the function has the representation ( 1 ). The classical difference formula with two nodes for the derivative has the form: u′(x)

L′2(u; x) = un un−1 ; xn−1 h x xn: Let u(x) = e−x=": Then with " = h it will be "j(u1 u0)=h u′(0)j = e−1: The relative error of the formula ( 3 ) does not decrease with mesh step decreasing. We need to develop formulas for numerical differentiation with accuracy that is uniform with respect to the parameter ".

By C and Cj we mean positive constants independent of the parameter " and the number of mesh nodes N: 2

Setting of the non-uniform

mesh

Let Ωh be a mesh on the interval [0; 1] :

We assume that the function u(x) of the form ( 1 ) is defined at the nodes of the mesh, un = u(xn); n = 0; 1; 2; : : : ; N:

Set Ωh as a Bakhvalov mesh [2] with the nodes xn = g(n=N ); n = 0; 1; : : : ; N; where the function g(t) is defined as follows: g(t) = 4" ln [1 2(1 ")t]; 0

t g(t) = + (2t 1)( 1 ); 1=2 1: For " e−1 we set the parameter = min

Given (7), we obtain that in the boundary layer region

It is easy to verify that the sequence of steps hn; n = 1; 2; : : : ; N=2 – is strictly increasing. From ( 8 ) it follows that Therefore, for some constant C2 there is an estimate: xn = 4" ln [1 2(1

")n=N ]; n = 0; 1; : : : ; xn = + (2n=N 1)(1

); n = N=2; : : : ; N: hn = 4" ln [1 + 1

]; n = 1; 2; : : : ; N=2: hN=2 = 4" ln [1 + 2(1 ") ]

:

N " C2

N hn ; n = 1; 2; : : : ; N: 3 Cubic spline on the Bakhvalov mesh On the constructed mesh Ωh we define a cubic spline S(u; x) 2 C2[0; 1] [8]

S(u; x) = (xn6hnx)3 Mn−1 + (x 6xhnn−1)3 Mn+

Proof. First, we estimate the error zn = Mn system:

NC2 (N " + 2) ln2 [1 + N2" ]: (12) u′′(xn). Given (10), we get that fzng is a solution of the hn 6 zn−1 + hn + hn+1 zn + hn+1 zn+1 = Fn; n = 1; 2; : : : ; N 3 6 1; z0 = 0; zN = 0; where

Fn = un+h1n+1 un un hnun−1 h6n u′n′−1 hn +3hn+1 u′n′ Given the Taylor series expansion with a remainder term in integral form, we obtain hn+1 u′n′+1: 6 Fn = x∫n+1[ xn

1 2hn+1 (xn+1 s)2 hn+1 ]u′′′(s) ds 6

Represent Fn in the form

Fn = Fn1

Fn2; Fn2 = Fn2 = Given that with u′′′(s) = const Fn1 = Fn2 = 0; we get

From (15) we obtain From (16) we get

xn 1 xn ∫ [ 1 (s

2hn xn−1)2

xn jFn2j < 23 h2n ∫ hjFn+n2j1 < 23 hn xn 1 xn ∫ ju( 4 )(s)j ds: Given the estimates (20)–(22) and the fact that "2jFn1j=hn+1 satisfies the same estimates, we obtain (18) (19) (20) (21) (22) (23) (24) (25) (26)

From (14) implies

Therefore,

"2 max jFnj n hn+1

= min { NC2 (N " + 2) ln [1 + N2" ]; NC }: We proceed to estimation of zn from (13). Divide the ratio (13) by hn+1 and get

hn 6hn+1 zn−1 + [ 1

+ 3

hn ] 3hn+1

1 zn + 6 zn+1 =

Fn ; n = 1; : : : ; N hn+1 1; z0 = 0; zN = 0: The matrix of the system (24) has a strict diagonal predominance in rows with the prevalence index 1=6; therefore, by the estimate (23) we get "2 mnax jMn u′′(xn)j 6 :

Let’s estimate the error in calculating of the second derivative at an arbitrary point x 2 [xn−1; xn]. It is easy to get

S′′(u; x) u′′(x) = zn−1 + (zn zn−1) x xn−1 + u′n′−1 + (u′n′ hn u′n′−1) x xn−1 hn u′′(x): Given an estimate (25) in (26) for "2zn and an estimate of the error of the linear interpolation formula for the function u′′(x) u′n′−1 + (u′n′ u′n′−1) x xn−1 hn u′′(x) hn

ju( 4 )(s)j ds ∫xn of the form (17), for some constant C we obtain "2jS′′(u; x) u′′(x)j

C ; x 2 [xn−1; xn]; n

N=2: It proves the estimate (11) for x 2 [xn−1; xn]; n N=2:

When x 2 [xn−1; xn]; n > N=2 an estimate (11) is correct, because the derivatives of the function u(x) to the fourth order are "-uniformly bounded.

Now let’s get the estimate of error in the calculation of the first derivative.

N=2: Given the representation ( 1 ) with constraints ( 2 ), for some constant C we get : Given (19) and the boundedness of the derivatives of the function u(x) outside the boundary layer region, for some constant C0 we get: "2 jFn2j hn+1 jFn2j hn+1

C0 N 2

; n < C0 N 2 ; n > "2 max jFnj n hn+1

NC2 (N " + 2) ln [1 + N2" ]: "2 mnax jFnj

Let z(x) = S(u; x) u(x); x 2 [xn−1; xn]: Due to the interpolation condition, there is s 2 (xn−1; xn) : z′(s) = 0: Then, by the mean value theorem, there is s0 : z′(x) z′(s) = z′′(s0)(x s): Given ( 8 ), (11) we obtain the estimate (12). The theorem is proved.

Remark. For " C=N from (11), (12) follows "jS′(u; x) u′(x)j ; "2jS′′(u; x) u′′(x)j for " = 1 error estimates coincide with known estimates in the regular case. 4

Results of numerical experiments Let us compare the accuracy in the calculation of derivatives based on spline interpolation when constructing a cubic spline on a uniform grid, Shishkin and Bakhvalov meshes.

Set the Shishkin mesh [1]: = min 2(1

N ) ; n > : According to [6], in the case of a function of the form ( 1 ) and the Shishkin mesh for some constant C the following error estimates hold "j ju(j)(x)

S(j)(x; u)j

ln4−j N C N 4−j ; x 2 [0; 1]; j = 1; 2: (27) when calculating the first derivative of the function u(x) in cases of a uniform mesh, Shishkin mesh and Bakhvalov mesh. Here x˜n;j are nodes of the condensed mesh, obtained from the division of each mesh interval [xn−1; xn] "

Acknowledgements The reported work of Zadorin N.A. was funded by RFBR, project number 19-31-60009. The work of Zadorin A.I. was funded by program of fundamental scientific researches of the SB RAS 1.1.3., project 0314-2019-0009.

References

[1] G.I. Shishkin Mesh Approximations of Singularly Perturbed Elliptic and Parabolic Equations . YekaterinburgUral. Otd. Ross. Akad. Nauk , 1992 (in Russian).

[2] N.S. Bakhvalov The optimization of methods of solving boundary value problems with a boundary layer . USSR Comput. Math. Math. Phys. 9 ( 4 ): 139 - 166 , 1969 .

[3] A.I. Zadorin , N.A. Zadorin Spline interpolation on a uniform mesh for functions with a boundary-layer component . Comput. Math. Math. Phys . 50 ( 2 ): 211 - 223 , 2010 .

[4] I.A. Blatov , A.I. Zadorin , E.V. Kitaeva On interpolation by cubic splines of the functions with a boundary layers . CEUR Workshop Proceedings . 1638 : 515 - 520 , 2016 .

[5] I.A. Blatov , A.I. Zadorin , E.V. Kitaeva Cubic spline interpolation of functions with high gradients in boundary layers . Comput. Math. Math. Phys . 57 ( 1 ): 7 - 25 , 2017 .

[6] I.A. Blatov , A.I. Zadorin , E.V. Kitaeva Approximation of a function and its derivatives on the basis of cubic spline interpolation in the presence of a boundary layer . Comput. Math. Math. Phys . 59 ( 3 ): 343 - 354 , 2019 .

[7] I.A. Blatov , A.I. Zadorin , E.V. Kitaeva On the parameter-uniform convergence of exponential spline interpolation in the presence of a boundary layer . Comput. Math. Math. Phys . 58 ( 3 ): 348 - 363 , 2018 .

[8] Yu.S. Zavyalov , B.N. Kvasov , V. L. Miroshnichenko Methods of Spline Functions . Moscow-Nauka, 1981 (in Russian).