Design of Numerical Algorithms for the Ballistic

ISSN 09655425, Computational Mathematics and Mathematical Physics, 2010, Vol. 50, No. 1, pp. 180–200. © Pleiades Publishing, Ltd., 2010. Original Russian Text © A.M. Blokhin, A.S. Ibragimova, B.V. Semisalov, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 1, pp. 188–208.

Design of Numerical Algorithms for the Ballistic Diode Problem A. M. Blokhina, A. S. Ibragimovab, and B. V. Semisalovb a

Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090 Russia b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia email: [email protected], [email protected], [email protected] Received April 23, 2009

Abstract—Numerical algorithms for finding stationary solutions to a hydrodynamic model of charge transport in semiconductors are proposed and described in detail. DOI: 10.1134/S0965542510010148 Key words: hydrodynamic model, ballistic diode, stabilization method, nonstationary regularization, predictor–corrector scheme, cubic interpolation spline, orthogonal sweep method.

INTRODUCTION At present, numerous mathematical models are available that describe, to a certain degree of reliability, physical phenomena in semiconductor devices. This motivates the task of designing and substantiating numerical algorithms for finding approximate solutions to such models. In this paper, we consider the bal listic diode problem, which is well known in the physics of semiconductors, and propose computational algorithms based on ideas other than those underlying usual finitedifference schemes. As the basic math ematical model, we use a recently proposed hydrodynamic model of charge transport in semiconductors (see [1, 2]). This model represents a quasilinear system of equations written in the form of conservation laws. The conservation laws are obtained from a system of moment relations for the Boltzmann transport equation by applying a certain truncation procedure (due to the variety of truncation procedures, there are numerous mathematical models describing charge transport in semiconductor devices). Several approaches are proposed below for finding approximate solutions. Specifically, we use a reduc tion of the problem to integral equations, spline functions and the predictor–corrector scheme, and orthogonal sweep method. The last algorithm yields results with guaranteed accuracy. This paper is organized as follows. The statement of the problem is presented in Section 1. The original mathematical model in the stationary case is discussed in Section 2. Section 3 describes a nonstationary regularization of the boundary value problem. The resulting initial boundary value problem is reduced to a system of integral equations in Section 4, while other computational algorithms are described in Section 5. Numerical results are addressed in the Appendix. 1. PRELIMINARIES In the onedimensional case, the dimensionless quasilinear nonstationary system of moment equa tions (for the nondimensionalizing procedure, see [3, 4]) is written as R t + J x = 0, 2 J t + ⎛  RE⎞ = RQ + c 11 J + c 12 I, ⎝3 ⎠ x ( RE ) t + I x = JQ + cP,

(1.1)

5 10 2 I t + ⎛ RE ⎞ =  REQ + c 21 J + c 22 I, ⎝9 ⎠x 3 εϕ xx = R – ρ. 180

(1.2)

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Here, R is the electron density, J = Ru, I = Rq, u is the electron velocity, q is the energy flux, E is the elec tron energy, P = Rσ, σ = 2 E – 1, Q = ϕx, ϕ is the electric potential, and ρ = ρ(x) is the doping density (a 3 given function on the interval [0, 1]). The coefficients c11, …, c22, c in system (1.1) are smooth functions of E. Their expressions can be found, for example, in [3, 4]. For system (1.1), (1.2), we state the ballistic diode problem, which is well known in the physics of semi conductors. This is a onedimensional problem describing a semiconductor device divided into three parts. The first and third are known as n+ regions, since they are characterized by a high doping concen tration. The middle part (channel) is a region with a low doping concentration (known as an n region). The width of the n+ – n – n+ channel, the doping density of the n+ region, and the doping density at the point x are used as the characteristic values L, N+ and N(x) = N, respectively (see [3, 4] and the Appendix). Then the dimensionless doping density ρ = N/N+ is such that 1 ≥ ρ ( x ) ≥ δ > 0,

x ∈ [ 0, 1 ].

A typical smoothed profile of ρ(x) is shown in Fig. 1. For system (1.1), the boundary conditions set at x = 0, 1 (t > 0) correspond to the ballistic diode prob lem (see [6–8]): R ( t, 0 ) = R ( t, 1 ) = 1, 3 E ( t, 0 ) = E ( t, 1 ) =  . 2

(1.3)

At t = 0, 0 < x < 1, we define the initial data R ( 0, x ) = R 0 ( x ),

J ( 0, x ) = J 0 ( x ),

E ( 0, x ) = E 0 ( x ),

I ( 0, x ) = I 0 ( x ),

(1.4)

where R0(x), E0(x) > 0. For Poisson equation (1.2), the boundary conditions at x = 0, 1 (t > 0) are specified as ϕ ( t, 0 ) = 0,

ϕ ( t, 1 ) = b˜ ,

(1.5)

where b˜ > 0 is the bias across the diode. A solution to system (1.1) and Eq. (1.2) is sought in the domain t > 0, 0 < x < 1. Treating Poisson’s equation (1.2) as an ordinary differential equation (with parameter t) for ϕ with the boundary conditions (1.5), we obtain (see [6]) 1

ϕ = ϕ ( t, x ) = b˜ x + β G ( x, s ) [ R ( t, s ) – ρ ( s ) ] ds,

∫ 0

β = 1 , ε

(1.6)

where G(x, s) is the Green’s function ⎧ s ( x – 1 ), G ( x, s ) = ⎨ ⎩ x ( s – 1 ),

0 0, system (1.8) is thyperbolic (strictly hyperbolic, see [9]). Note that, by virtue of (1.9), the number of boundary conditions (see (1.3)) specified on each of the boundaries x = 0, 1 coincides with the number of ontgoing characteristics for each boundary (two in each case). Problem (1.1)–(1.5) with b˜ = 0 has the stationary solution (global thermodynamic equilibrium) J ( t, x ) ≡ ˆJ = 0,

ˆ = 3 , E ( t, x ) ≡ E 2

I ( t, x ) ≡ ˆI = 0,

(1.10) ˆ (x) ϕ ˆ ˆ ( x ), R ( t, x ) = R ( x ) = e , ϕ ( t, x ) = ϕ ˆ (x) satisfies the Poisson equation where ϕ ˆ = e ϕˆ – ρ εϕ'' (1.11) with the boundary conditions ˆ (0) = ϕ ˆ ( 1 ) = 0. ϕ (1.12) Note that ε is sufficiently small for actual semiconductor devices (see [3, 4, 6]). The boundary value problem (1.11), (1.12) was investigated in detail in [10], where the following result was proved. Theorem 1.1. Let ρ = ρ(x) ∈ C2[0, 1], 0 < δ ≤ ρ(x) ≤ 1, and let (ρ(x) – 1) be compactly supported (see Fig. 1). Then there is ε0 > 0 such that, for any ε with 0 < ε ≤ ε0, the boundary value problem (1.11), (1.12) has a unique solution of the form ˆ ( x ) = ln ρ ( x ) + O ( ε 1/2 ). ϕ (1.13) Here, C2[0, 1] is the space of smooth functions equipped with the norm ρ

2

C [ 0, 1 ]

= max ρ ( x ) + max ρ' ( x ) + max ρ'' ( x ) . x ∈ [ 0, 1 ]

x ∈ [ 0, 1 ]

x ∈ [ 0, 1 ]

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Note that ε0 depends on δ. The proof of this theorem is constructive: an explicit expression for the sec ond term in (1.13) is given in [10]. Finally, we note that, if ρ ( x ) ≡ 1, then problem (1.11), (1.12) has the unique solution ˆ ( x ) ≡ 0. ϕ 2. PROBLEM (1.1)–(1.5) IN THE STATIONARY CASE Below, we describe computational algorithms for finding stationary solutions of problem (1.1)–(1.5). For this purpose, consider Eqs. (1.1) and (1.2) in the stationary case: dJ/dx = 0, (2.1) dI/dx = JQ + cP, (2.2) d 2  ⎛  RE⎞ = RQ + c 11 J + c 12 I, (2.3) dx ⎝ 3 ⎠ 2 d ⎛ 4 RE ⎞ = 2 REQ + 2 c 21 J + 2 c 22 I, ⎝ ⎠ dx 9 5 3 5 2

(2.4)

2

d ϕ/dx = β ( R – ρ ), β = 1/ε. Equations (2.1) and (2.2) are rewritten as du/dx = – uX, χ = ln R, X = dχ/dx, dq/dx = – qX + uQ + cσ. After simple transformations, from (2.3) and (2.4), we obtain Σ = au + bq, Σ = dσ/dx, Q . Χ = mu + nq +  1+σ Here, c 11 – a 2 c 21 2 c 22 a = a(E) =    – c11, b = b(E) =    – c12, m = m(E) =  , 51 + σ 51 + σ 1+σ Regarding (2.8) and (2.9) as a system for determining u and q, we easily find u = F ( E ) { Q – ( 1 + σ )X – F 0 ( E )Σ },

(2.5) (2.6) (2.7) (2.8) (2.9) c 12 – b n = n(E) =  . 1+σ (2.10)

q = G ( E ) { – Q + ( 1 + σ )X + G 0 ( E )Σ },

(2.11)

where 5 c 22 –  Ec 12 3 F ( E ) = – , det 5 Ec 12 3 F 0 ( E ) = 1 – , c 22 – 5 Ec 12 3

5 c 21 –  Ec 11 3 G ( E ) = – , det 5 Ec 11 3 G 0 ( E ) = 1 – , c 21 – 5 Ec 11 3

det = c 11 c 22 – c 21 c 12 . Applying the operator d/dx to (2.8) and (2.9) and using (2.6), (2.7), (2.10), and (2.11) gives d σ/dx = Ᏺ 2

2

(σ)

2

2

( Σ, X, Q, σ ) = a 1 Σ + a 2 ΣX + a 3 ΣQ + a 4 XQ + a 5 Q + bcσ, (χ)

(2.12)

d χ/dx = Ᏺ ( Σ, X, Q, σ, χ, ρ ) = – X + b 1 Σ + b 2 ΣX 2

2

2

2

(2.13)

2 β  ( e χ – ρ ) + ncσ. + b 3 ΣQ + b 4 XQ + b 5 Q +  1+σ

Here, a1 = –a'F(E)F0(E) + b'G(E)G0(E),

a2 = –1 + (1 + σ){b'G(E) – a'F(E)},

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a3 = a'F(E) – b'G(E) – bF(E)F0(E), a4 = –b(1 + σ)F(E), a5 = bF(E), b1 = –m'F(E)F0(E) + n'G(E)G0(E), b2 = (1 + σ){n'G(E) – m'F(E)}, 1 1 b3 = – 2 + m'F(E) – n'G(E) – nF(E)F0(E), b4 =  + bF(E)F0(E), b5 = nF(E), 1+σ (1 + σ) da 3 da a' =  =   , dσ 2 dE

etc.

Equation (2.5) is rewritten as (ϕ)

χ

d ϕ/dx = Ᏺ ( χ, ρ ) = β ( e – ρ ). (2.14) Thus, in the stationary case, the original problem (1.1)–(1.5) can be reduced to an initial boundary value problem for system (2.12)–(2.14). The boundary conditions for this system follow from (1.3) and (1.5): χ = σ = 0 at x = 0, 1, 2

2

ϕ = 0 ϕ = b˜

x = 0,

at

(2.15)

at x = 1. After σ, χ, and ϕ have been determined, u and q can be found by formulas (2.10) and (2.11). 3. NONSTATIONARY REGULARIZATION OF BOUNDARY VALUE PROBLEM (2.12)–(2.15) To solve problem (2.12)–(2.15), we use the stabilization method (the general idea behind the method is described, for example, in [11]). For this purpose, problem (2.12)–(2.15) is replaced by its nontrivial non stationary regularization (the application of a similar regularization in numerical computations can be found, for example, in [12, 13]). More specifically, Eqs. (2.12)–(2.14) are replaced by the relations (ϕ)

( 1 – ξ )τϕ = ξ ϕ – Ᏺ ( χ, ρ ), 2

2

(3.1)

(σ)

( 1 – ξ )τσ = ξ σ – Ᏺ ( ξσ, ξχ, ξϕ, σ ), 2

2

(3.2)

(χ)

( 1 – ξ )τχ = ξ χ – Ᏺ ( ξσ, ξχ, ξϕ, σ, χ, ρ ). (3.3) Here, τ = ∂/∂t and ξ = ∂/∂x are differential operators and t plays the role of time. A solution of system (3.1)–(3.3) is sought in the domain t > 0, 0 < x < 1. At t = 0, we set the initial data (see (1.4)) ϕ ( 0, x ) = ϕ 0 ( x ), 2

2

2 (3.4) σ ( 0, x ) = σ 0 ( x ) =  E 0 ( x ) – 1, 3 χ ( 0, x ) = χ 0 ( x ) = ln R 0 , 0 < x < 1. At x = 0, 1, t > 0, the unknown functions ϕ(t, x), σ(t, x), and χ(t, x) satisfy the boundary conditions (see (2.15)) χ = σ = 0 at x = 0, 1, ϕ = 0 at x = 0, (3.5) ϕ = b˜ at x = 1. Remark 3.1. Making the substitution ˜ + b˜ x, ϕ = ϕ ˜ (t, x) is a new dependent variable, we obtain homogeneous boundary conditions for ϕ ˜: where ϕ ˜ = 0 ϕ

at

x = 0, 1.

˜ + b˜ (in what follows, the tilde over ϕ is omitted). Moreover, ξϕ in Eqs. (3.2) and (3.3) is replaced by ξϕ Remark 3.2. Below, we also need the following form of system (3.1), (3.2):

˜ (ϕ)

τθ ϕ + θ ϕ = Ᏺ , COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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˜ (σ)

τθ σ + θ σ = Ᏺ ,

(3.2)'

˜ (χ)

τθ χ + θ χ = Ᏺ .

(3.3)'

Here, 2

θ ϕ = ϕ – ξ ϕ,

2

θ σ = σ – ξ σ,

2

θ χ = χ – ξ χ,

˜ (ϕ) = ϕ – Ᏺ(ϕ) , Ᏺ ˜ (σ) = σ – Ᏺ(σ) , Ᏺ ˜ (χ) = χ – Ᏺ(χ) . Ᏺ To substantiate the stabilization method, we obtain a global a priori estimate for the initial boundary value problem (3.1)–(3.5). Unfortunately, this a priori estimate is derived under certain constraints on b˜ and ρ(x) (the general case still remains an open problem). Assume that problem (3.1)–(3.5) has a suffi ciently smooth solution on the interval [0, t∗], where t∗ > 0 is a constant. Define the constant M * = max { max σ ( t ) t ∈ [ 0, t ] *

where σ ( t )

⺓ [ 0, 1 ]

⺓ [ 0, 1 ],

max χ ( t )

t ∈ [ 0, t ] *

⺓ [ 0, 1 ] },

= max σ ( t, x ) , etc. x ∈ [ 0, 1 ]

To simplify the subsequent presentation, we set b˜ = 0 and ρ(x) ≡ 1 (the technique used below to derive = the a priori estimate can be extended to the case when the constants b˜ > 0 and ||ρ – 1|| ⺓[0, 1]

max ρ ( x ) – 1 = (1 – δ) > 0 are small). Assume that the solution of the initial boundary value problem (3.1)–

x ∈ [ 0, 1 ]

ˆ ≡ 0, see (3.5) on [0, t∗] differs little from equilibrium (1.10) (in view of b˜ = 0 and ρ(x) ≡ 1, we then have ϕ Section 1). By virtue of what was said above, the constant M∗ is also small. Multiplying Eq. (3.1) by 2ϕ, Eq. (3.2) by 2σ, Eq. (3.3) by 2χ, Eq. (3.1)' by 2θϕ, Eq. (3.2)' by 2θσ, and Eq. (3.3)' by 2θϕ, after simple rearrangements, we obtain 2

2

χ

2

τ { ϕ + ( ξϕ ) } – 2ξ ( ϕτξϕ ) – 2ξ ( ϕξϕ ) + 2 ( ξϕ ) = 2ϕβ ( 1 – e ), 2

2

2

τ { σ + ( ξσ ) } – 2ξ ( στξσ ) – 2ξ ( σξσ ) + 2 ( ξσ ) + 2bcσ 2

(3.6)

2

(3.7)

2

= 2σ { a 1 ( ξσ ) + a 2 ξσξχ + a 3 ξσξϕ + a 4 ξχξϕ + a 5 ( ξϕ ) }, 2

2

2

τ { χ + ( ξχ ) } – 2ξ ( χτξχ ) – 2ξ ( χξχ ) + 2 ( ξχ ) + 2ncχσ 2βχ ( e χ – 1 ) = – 2χ { – ( ξχ ) 2 + b ( ξσ ) 2 + b ξσξχ + b ξσξϕ + b ξχξϕ + b ( ξϕ ) 2 }, +  1 2 3 4 5 1+σ 2

χ

2

τθ ϕ + 2θ ϕ – 2θ ϕ ϕ = 2θ ϕ β ( 1 – e ), 2

2

(3.8)

(3.9)

2

2

τθ σ + 2θ σ – 2θ σ σ + 2bcθ σ σ = – 2θ δ { a 1 ( ξσ ) + a 2 ξσξχ + a 3 ξσξϕ + a 4 ξχξϕ + a 5 ( ξϕ ) },

(3.10)

2βθ χ 2 2 τθ χ + 2θ χ – 2θ χ χ + 2ncθ χ σ + χ ( e – 1 ) 1+σ

(3.11)

2

2

2

= –2θ χ { – ( ξχ ) + b 1 ( ξσ ) + b 2 ξσξχ + b 3 ξσξϕ + b 4 ξχξϕ + b 5 ( ξϕ ) }. Integrating (3.6) and (3.7) with respect to x from 0 to 1 with the boundary conditions (3.5) taken into account and adding up the resulting expressions multiplied by positive constants α1, …, α6 gives τΦ ( t ) + Φ 1 ( t ) = Φ 2 ( t ) + Φ 3 ( t ).

(3.12)

Here, 1

Φ(t) =

∫ {α (ϕ 1

2

2

2

2

2

2

2

2

2

No. 1

2010

+ ( ξϕ ) ) + α 2 ( σ + ( ξσ ) ) + α 3 ( χ + ( ξχ ) ) + α 4 θ ϕ + α 5 θ σ + α 6 θ χ } dx,

0

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⎧ βχα χ 2 2 2 2 Φ 1 ( t ) = 2 ⎨ α 1 ( ξϕ ) + α 2 ( ξσ ) + α 2 bcσ + α 3 ( ξχ ) + α 3 ncχσ + 3 ( e – 1 ) 1+σ ⎩ 0

∫

⎫ 2 2 2  + α 4 θ ϕ – α 4 θ ϕ ϕ + α 5 θ σ – α 5 θ σ σ ( 1 – bc ) + α 6 θ χ – α 6 θ χ χ + α 6 ncθ χ σ ⎬dx, ⎭ 1

⎧ βθ χ χ χ χ ⎫ Φ 2 ( t ) = 2 ⎨ α 1 ϕβ ( 1 – e ) + α 4 θ ϕ β ( 1 – e ) +  α 6 ( 1 – e ) ⎬dx, 1+σ ⎩ ⎭ 0

∫

1

∫

2

2

Φ 3 ( t ) = – 2 { ( α 2 σ + α 5 θ σ ) [ a 1 ( ξσ ) + a 2 ξσξχ + a 3 ξσξϕ + a 4 ξχξϕ + a 5 ( ξϕ ) ] 0

2

2

2

+ ( α 3 χ + α 6 θ χ ) [ – ( ξχ ) + b 1 ( ξσ ) + b 2 ξσξχ + b 3 ξσξϕ + b 4 ξχξϕ + b 5 ( ξϕ ) ] }dx. Below, we need the following relations: (i) the Poincaré inequality (see [14]) for ϕ, σ, χ 1

1

∫ ( ξϕ ) dx ≥ 2 ∫ ϕ dx 2

2

0

0

(and similar inequalities for σ and χ); (ii) the inequality χ(eχ – 1) ≥ 0, which holds for any continuous functions χ; (iii) the simplest embedding theorem (see [14–16]): for σ and χ, 1

1/2

⎛ 2 ⎞ σ ( t ) ⺓ [ 0, 1 ] ≤ ⎜ ( ξσ ) dx⎟ ⎝ ⎠

∫

,

0

and, for ϕ, σ, and χ, ϕ(t)

1

⺓ [ 0, 1 ]

≤ Mb ϕ ( t )

2

W 2 ( 0, 1 )

,

where ϕ(t)

1

⺓ [ 0, 1 ]

= max ϕ ( t, x ) + max ξϕ ( t, x ) , x ∈ [ 0, 1 ]

x ∈ [ 0, 1 ]

1

ϕ(t)

2

W 2 ( 0, 1 )

⎛ ⎞ 2 2 2 2 = ⎜ { ϕ ( t, x ) + ( ξϕ ( t, x ) ) + ( ξ ϕ ( t, x ) ) } dx⎟ ⎝ ⎠

1/2

∫ 0

2

is the norm of ϕ(t, x) in the Sobolev space W 2 (0, 1) (similar inequalities for σ, χ), and Mb > 0 is an embed ding constant (see [15]); (iv) estimates of the form 1

1

1

∫ a σ ( ξσ ) dx ≤ ∫ a 2

1

0

2

1

σ ( ξσ ) dx ≤ M a1 σ ( t )

⺓ [ 0, 1 ]

0

0

1

1

∫

∫

⎞ ⎧⎛ 2 2 a 2 σξσξχ dx ≤ M a2 ⎨ ⎜ ( ξσ ) dx⎟ ⎠ ⎩⎝ 0

3/2

0

1

∫ 0

∫

1

∫

1

⎛ ⎞ 2 + ( ξχ ) dx ⎜ ( ξσ ) dx⎟ ⎝ ⎠

∫

3/2

,

0

1

2

0

⎞ 2⎛ 2 2 a 1 θ δ ( ξσ ) dx ≤ M a1 M b ⎜ θ σ dx⎟ ⎝ ⎠

∫

1

⎛ 2 ⎞ ( ξσ ) dx ≤ M a1 ⎜ ( ξσ ) dx⎟ ⎝ ⎠ 2

∫ 0

1/2

⎫ ⎬, ⎭

1/2

σ(t)

2

W 2 ( 0, 1 )

,

0

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where M a1 and M a2 are positive constants depending on M∗; (v) the Cauchy inequality with ε˜ : 2 1 2 aˆ bˆ ≤ aˆ + ε˜ bˆ 4ε˜

for any aˆ , bˆ , ε˜ > 0; (vi) the inequality |eχ – 1| ≤ |χ|e|χ|, which holds for any continuous function χ. Taking into account inequalities (i)–(vi), we obtain 1

Φ 1 ( t ) – Φ 2 ( t ) ≥ { α 1 ( ξϕ ) + α 2 ( ξσ ) + α 3 ( ξχ ) + 2 ( Y, Ꮽ Y ) } ≥ M 1 Φ ( t ),

∫

2

2

2

(3.13)

0

where ⎛ ⎜ α 1 ( 1 – βε˜ ) ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 Ꮽ = ⎜⎜ ⎜ – 1 α 4 ⎜ 2 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎝ ⎛ ⎜ ⎜ ⎜ Y = ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ θσ ⎟ ⎟ θχ ⎠ ϕ σ χ θϕ

⎞ ⎟ ⎟ ⎟ α 2 ( 1 + bc ) 1 α 3 nc 0 – 1 α 5 ( 1 – bc ) 1 α 6 nc ⎟ 2 2 2 ⎟ ⎟ 1  α 3 nc A 0 0 – 1 α 6 ⎟ 2 2 ⎟, ⎟ 0 0 α 4 ( 1 – βε˜ ) 0 0 ⎟ ⎟ ⎟ 1 –  α 5 ( 1 – bc ) 0 0 α5 0 ⎟ ⎟ 2 ⎟ 1 α 6 nc – 1 α 6 0 0 α6 z ⎟ ⎠ 2 2 0

βε˜ ⎞ z = ⎛ 1 –  , ⎝ 1 – M *⎠

0

1 –  α4 2

0

0

α 6 ⎞ e 2M *   β, A = α 3 – ⎛ α 1 + α 4 +  ⎝ 1 – M *⎠ 4ε˜

M1 and ε˜ are positive constants; and we obtain 3/2

Φ3 ( t ) ≤ M2 ( Φ ( t ) ) ,

(3.14)

where M2 is a positive constant. 2 By choosing α1, …, α6 such that α1 = α4 = α6 = ε˜ , α3 = 1, α5 = ε˜ , and α2 = 1/ ε˜ , where ε˜ > 0 is suffi ciently small, is easy to see that inequality (3.13) holds true (for these α1, …, α6, Ꮽ > 0), and the constant M1 > 0 in (3.13) is eventually determined in terms of the constants ε˜ and M∗. Note also that, at the equi ˆ = 3/2 (see Section 1), we have b( E ˆ )c( E ˆ ) > 0 (see [3, 4]). The constant M in (3.14) depends librium E = E

2

on ε˜ , M∗, and Mb. The rest of the derivation of the a priori estimate is fairly standard (see, e.g., [5]). Specifically, in view of (3.13) and (3.14), relation (3.12) yields 3/2

τΦ ( t ) + M 1 Φ ( t ) ≤ M 2 ( Φ ( t ) ) .

(3.15)

We have

Ᏺ ( Φ ( t ) ) = – M1 Φ ( t ) + M2 ( Φ ( t ) )

3/2

< 0,

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if 2

0 < Φ ( t ) < ( M 1 /M 2 ) . Therefore, (3.15) implies, as a matter of fact, local a priori estimate – νt

Φ ( t ) ≤ e Φ ( 0 ), 0 < t ≤ t * , (3.16) where ν (0 < ν < M1) is a sufficiently small constant. Remark 3.3. Estimate (3.16) can be used to prove a local existence theorem for a sufficiently smooth solution of the initial boundary value problem (3.1)–(3.5). By the local existence theorem and a priori estimate (3.16), a smooth solution can be extended to the entire time interval t > 0. Indeed, if initial data (3.4) satisfy 2

0 < Φ ( 0 ) < ( M 1 /M 2 ) , then (3.16) implies that Φ(t∗) ≤ Φ(0). Therefore, σ(t*)

≤ Mb σ ( t * )

⺓ [ 0, 1 ]

2

W 2 ( 0, 1 )

≤ Mb M3 ( Φ ( t * ) )

1/2

1/2

≤ Mb M3 ( Φ ( 0 ) ) ,

(3.17)

where M3 > 0 is a constant determined eventually in terms of ε˜ and M∗. Choosing initial data that are suf ficiently close to the equilibrium, we find from (3.17) that ||σ(t∗)||⺓[0, 1] ≤ M∗, etc. Thus, if the initial data are specified by ϕ(t∗, x), σ(t∗, x), and χ(t∗, x), then the smooth solution of the initial boundary value problem (3.1)–(3.5) can be extended to the interval [t∗, 2t∗] so that estimate (3.16) holds: Φ(t) ≤ e

–ν ( t – t * )

– νt

Φ ( t * ) ≤ e Φ ( 0 ).

Thus, (3.16) is a global a priori estimate, which implies the nonlinear asymptotic (Lyapunov) stability of equilibrium (1.10). This means that we have substantiated the stabilization method (at the differential level) at least in the special case of b˜ = 0 and ρ(x) ≡ 1. 4. REDUCTION OF THE INITIAL BOUNDARY VALUE PROBLEM (3.1)–(3.5) TO A SYSTEM OF INTEGRAL EQUATIONS In view of the different form of system (3.1)–(3.3) presented in Section 3 (see Eqs. (3.1)'–(3.3)'), solutions of the the initial boundary value problem (3.1)–(3.5) can be found in a different manner (see about it [17]). Treating Eqs. (3.1)'–(3.3)' as ordinary differential equations (with parameter x), we easily obtain t –t

∫

θ ϕ ( t, x ) = e θ ϕ0 ( x ) + e

–( t – ζ )

(ϕ)

{ ϕ ( ζ, x ) – Ᏺ ( χ ( ζ, x ), ρ ( x ) ) } dζ,

(4.1)

0

t –t

∫

θ σ ( t, x ) = e θ σ0 ( x ) + e

–( t – ζ )

{ σ ( ζ, x ) – Ᏺ

(σ)

( ξσ ( ζ, x ), ξχ ( ζ, x ), ξϕ ( ζ, x ) + b˜, σ ( ζ, x ) ) } dζ,

(4.2)

0

–t

θ χ ( t, x ) = e θ χ0 ( x ) t

∫

+ e

–( t – ζ )

(χ) { χ ( ζ, x ) – Ᏺ ( ξσ ( ζ, x ), ξχ ( ζ, x ), ξϕ ( ζ, x ) + b˜, σ ( ζ, x ), χ ( ζ, x ), ρ ( x ) ) } dζ,

(4.3)

0

where θ ϕ0 (x) = ϕ0(x) – ξ2ϕ0(x), θ σ0 (x) = σ0(x) – ξ2σ0(x), θ χ0 (x) = χ0(x) – ξ2χ0(x), and ϕ0(x), σ0(x), and χ0(x) are initial data (see (3.4) in Section 3). Supplementing the relations 2

ξ ϕ – ϕ = – θϕ ,

2

ξ σ – σ = – θσ ,

2

ξ χ – χ = –θχ

with conditions (3.5) (see Section 3), we obtain the following boundary value problems (with parameter t): COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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2

0 < x < 1,

ϕ ( t, 0 ) = ϕ ( t, 1 ) = 0,

(4.4)

2

0 < x < 1,

σ ( t, 0 ) = σ ( t, 1 ) = 0,

(4.5)

ξ χ – χ = – θ χ , 0 < x < 1, χ ( t, 0 ) = χ ( t, 1 ) = 0. The solution of problems (4.4)–(4.6) is conveniently written as

(4.6)

ξ ϕ – ϕ = –θϕ , ξ σ – σ = –θσ , 2

1

∫

ϕ ( t, x ) = – G ( x, s )θ ϕ ( t, s ) ds,

(4.7)

0 1

∫

σ ( t, x ) = – G ( x, s )θ σ ( t, s ) ds,

(4.8)

0

1

∫

χ ( t, x ) = – G ( x, s )θ χ ( t, s ) ds.

(4.9)

0

Here, G(x, s) is the Green’s function: ⎧ sinh s sinh ( x – 1 ) ⎪  , ⎪ sinh 1 G ( x, s ) = ⎨ sinh x sinh ( s – 1) , ⎪  ⎪ sinh 1 ⎩

0 0,

0 < x < 1,

ϕ = 0 at x = 0, 1, t > 0, ϕ = ϕ 0 ( x ) at t = 0, 0 < x < 1.

(5.1)

Here, f(x) is a given function. To discretize the equation (1 – ξ2)τϕ = ξ2ϕ – f(x) in t, we introduce the following notation: ϕn(x) = ϕ(nΔ, x) = ϕ, n = 0, 1, …, and Φ = ϕn + 1(x), Δ is the mesh size in t. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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Approximating the derivative τϕ by the expression (Φ – ϕ)/Δ produces the equation ξ Φ = ᏮΦ + Ᏺ, 2

(5.2)

where 2

1 , Ᏺ = ξ ϕ – ϕ + Δf . Ꮾ =  1+Δ

1+Δ

2

An approximate solution is sought in the form of an interpolation cubic spline of the class ⺓ (see [18]): 2

h S ( x ) = ( 1 – τ )Φ k + τΦ k + 1 –  τ ( 1 – τ ) [ ( 2 – τ )m k + ( 1 + τ )m k + 1 ], 6

(5.3)

where x–x τ = k , h x k = kh,

x ∈ [ x k, x k + 1 ],

k = 0, K – 1 ,

Φ k = Φ ( x k ),

Kh = 1, 2

m k = ξ Φ ( x k ).

The cubic spline (5.3) must be continuous, together with its first derivative, everywhere on [0, 1] (the second derivative is continuous by the definition of spline (5.3)). The first derivative of the cubic spline is calculated as Φk + 1 – Φk h 2 2 ξS ( x ) =   –  [ ( 2 – 6τ + 3τ )m k + ( 1 – 3τ )m k + 1 ], h 6

x ∈ [ x k, x k + 1 ],

k = 0, K – 1 .

Then, calculating the derivatives ξS ( x k + 0 ),

ξS ( x k – 0 ),

where Φk + 1 – Φk h ξS ( x k + 0 ) =   –  [ ( 2m k + m k + 1 ) ], h 6 Φk – Φk – 1 h ξS ( x k – 0 ) =   +  [ ( m k – 1 + 2m k ) ], h 6 and equating them, we obtain 1 m k – 1 + 2m k + 1 m k + 1 = 32 ( ηΦ k – ηΦ k ), 2 2 h

k = 1, K – 1 .

(5.4)

Here, η = ψ – 1 and η = 1 – ψ–1 are difference operators and ψ and ψ–1 are shift operators: ψ±1Φk = Φk ± 1 (ψ+1 = ψ). Setting x = xk in (5.2) and substituting ξ2Φ(xk) into (5.4) gives the threelevel difference scheme 2 2 2 2 ⎧ ⎧ ⎧ h ⎫ h ⎫ h ⎫ h 1 – Ꮾ Φ – 2 1 + Ꮾ Φ + 1 – Ꮾ Φ = { Ᏺ k – 1 + 4 Ᏺ k + Ᏺ k + 1 },          ⎨ ⎬ k–1 ⎨ ⎬ k ⎨ ⎬ k+1 6 ⎭ 3 ⎭ 6 ⎭ 6 ⎩ ⎩ ⎩

k = 1, K – 1 , (5.5)

where Ᏺk = Ᏺ(xk). To determine the grid function Φk (k = 0, K ) from the system of algebraic equations (5.5), we have to set boundary conditions at k = 0 and k = K. It follows from (5.1) that Φ 0 = 0, Φ K = 0. (5.6) System (5.5), (5.6) can be solved by sweep method (see [11]). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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The subsequent reasoning concerning the application of the computational algorithm to problem (3.1)–(3.5) (σ) is obvious. We only make the following remark. The values of ξσ, ξχ, and ξϕ on the righthand sides Ᏺ (χ) and Ᏺ at grid points (i.e., at x = hk = xk, k = 0, K ) are calculated with the help of spline (5.3). Since Φk + 1 – Φk h 2 2 ξS ( x ) =   –  [ ( 2 – 6τ + 3τ )m k + ( 1 – 3τ )m k + 1 ], h 6

x ∈ [ x k, x k + 1 ],

k = 0, K – 1 ,

we have Φk + 1 – Φk h ξS ( x k + 1 ) =   +  [ m k + 2m k + 1 ], h 6

x ∈ [ x k, x k + 1 ],

k = 0, K – 1 ,

Φ1 – Φ0 h ξS ( 0 ) =   +  [ 2m 0 + m 1 ]. h 6 The terms Ᏺk in (5.5) are calculated according to the following scheme: for Eq. (3.1), Δ 1  ( ξ ϕ ( nΔ, x ) – ϕ ), Ᏺ k =  Ᏺ ( χ k , ρ k ) +  k k (ϕ)

2

n

1+Δ

n

1+Δ

n

χ k = χ ( nΔ, x k ),

ρ k = ρ ( x k ),

k = 0, K ;

for Eq. (3.2), Δ  Ᏺ ( ξσ ( nΔ, x ), ξχ ( nΔ, x ), ξϕ ( ( n + 1 )Δ, x ) + b˜, σ ) Ᏺ k =  k k k k (σ)

n

1+Δ

n 2 1 +  ( ξ σ ( nΔ, x k ) – σ k ), 1+Δ

k = 0, K ;

and, for Eq. (3.3), Δ Ᏺ k =   Ᏺ { ξσ ( ( n + 1 )Δ, x k ), ξχ ( nΔ, x k ), ξϕ ( ( n + 1 )Δ, x k ) + b˜, σ k , χ k , ρ k } (χ)

n+1

n

1+Δ

2 n 1 +  ( ξ χ ( nΔ, x k ) – χ k ), 1+Δ

k = 0, K .

5.2. Now, we briefly describe the computational algorithm proposed for the system of the integral equa tions (4.1)–(4.3). The computations are performed on a grid with the variable mesh size hk = xk + 1 – xk, k = 0, K – 1 . The n

n

integral equations are discretized in t, and the following notation is introduced: ( θ ϕ ) i = θϕ(nΔ, xi), ϕ i = ϕ(nΔ, xi), etc., where Δ is the mesh size in t. As a result, we obtain the relations 1 n ϕi

∫

= – G ( x i, s )θ ϕ ( ( n – 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξϕ ( nΔ, x i ) = – cosh ( x i – s )θ ϕ ( ( n – 1 )Δ, s ) ds – i sinh ( s – 1 )θ ϕ ( ( n – 1 )Δ, s ) ds, sinh 1 0

n

( θϕ )i = e

0

– nΔ ⎛

⎜ θ ϕ ( 0, x i ) + ⎝

nΔ

∫ 0

ζ

e { ϕ ( ζ, x i ) – Ᏺ

(ϕ)

⎞ ( χ ( ζ, x i ), ρ ( x i ) ) } dζ⎟ ; ⎠

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BLOKHIN et al. 1 n σi

∫

= – G ( x i, s )θ σ ( ( n – 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξσ ( nΔ, x i ) = – cosh ( x i – s )θ σ ( ( n – 1 )Δ, s ) ds – i sinh ( s – 1 )θ σ ( ( n – 1 )Δ, s ) ds, sinh 1 0

n

( θσ )i = e

(5.8)

0

nΔ

– nΔ ⎛

⎜ θ σ ( 0, x i ) + ⎝

∫ 0

⎞ (σ) ζ e { σ ( ζ, x i ) – Ᏺ ( ξσ ( ζ, x i ), ξχ ( ζ, x i ), ξϕ ( ζ, x i ) + b˜, σ ( ζ, x i ) ) } dζ⎟ ; ⎠ 1 n

∫

χ i = – G ( x i, s )θ χ ( ( n – 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξχ ( nΔ, x i ) = – cosh ( x i – s )θ χ ( ( n – 1 )Δ, s ) ds – i sinh ( s – 1 )θ χ ( ( n – 1 )Δ, s ) ds, sinh 1 0

0

n ( θχ )i

nΔ

+

ζ

∫ e { χ ( ζ, x ) – Ᏺ i

0

(χ)

= e

– nΔ ⎛

⎜ θ χ ( 0, x i ) ⎝

(5.9)

∫

⎞ ( ξσ ( ζ, x i ), ξχ ( ζ, x i ), ξϕ ( ζ, x i ) + b˜, σ ( ζ, x i ), χ ( ζ, x i ), ρ ( x i ) ) } dζ ⎟ , ⎠

where 2

θ ϕ ( 0, x i ) = ϕ 0 ( x i ) – ξ ϕ 0 ( x i ),

2

θ σ ( 0, x i ) = σ 0 ( x i ) – ξ σ 0 ( x i ), 2

θ χ ( 0, x i ) = χ 0 ( x i ) – ξ χ 0 ( x i ). The integrals in (5.7)–(5.9) can be evaluated using the trapezoidal, Simpson, or other method. 5.3. An approximate solution to the initial boundary value problem (3.1)–(3.5) can also be found by applying the predictor–corrector scheme. We need system (3.1)'–(3.3)' and the integral equations (4.7)– (4.12). For the equation

˜ (ϕ)

τθ ϕ + θ ϕ = Ᏺ , the predictor–corrector scheme is written as n+1

n

(5.10)

n

θϕ – θϕ n+1 ˜ (ϕ)   + θϕ = ( Ᏺ ) * . Δ

θ ϕ* – θ ϕ ˜ (ϕ) n   + θ ϕ* = ( Ᏺ ) , Δ/2 n+1

Expressing θ ϕ* from the first equation and θ ϕ

n Δ ˜ (ϕ) n θ ϕ +  ( Ᏺ ) 2 θ ϕ* =  , Δ 1 +  2

from the second yields n+1 θϕ

˜ (ϕ)

Δ(Ᏺ )* + θ = ϕ . 1+Δ n

As a result, we obtain

˜ ( ϕ ) ) n = ϕ n – Ᏺ ϕ ( χ n, ρ ), i i i i n ˜ (ϕ) n (θ ) + Δ  ( Ᏺ )

(Ᏺ

ϕ

i i 2 ( θ *i ) i =  , 1 + Δ/2

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1

∫

ϕ *i = – G ( x i, s )θ *ϕ ( s ) ds, 0

˜ ( ϕ ) ) * = ϕ * – Ᏺ ϕ ( χ n, ρ ), i i i i

(Ᏺ

n+1 ( θϕ )i

(5.11)

˜ (ϕ)

Δ ( Ᏺ ) *i + ( θ ϕ ) i =  , 1+Δ n

1 n+1

ϕi

∫

= – G ( x i, s )θ ϕ ( ( n + 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξϕ ( ( n + 1 )Δ, x i ) = – cosh ( x i – s )θ ϕ ( ( n + 1 )Δ, s ) ds – i sinh ( s – 1 )θ ϕ ( ( n + 1 )Δ, s ) ds; sinh 1 0

˜

(Ᏺ

(σ) n )i

0

σ n n = σ i – Ᏺ ( ξσ ( nΔ, x i ), ξχ ( nΔ, x i ), ξϕ ( ( n + 1 )Δ, x i ) + b˜, σ i ),

( θσ )i + Δ  ( Ᏺ ) i 2 ( θ *σ ) i =  , 1 + Δ/2 n

˜ (σ)

n

1

∫

σ i* = – G ( x i, s )θ σ* ( s ) ds, 0

xi

1

∫

∫

cosh x ξσ* ( x i ) = – cosh ( x i – s )θ σ* ( s ) ds – i sinh ( s – 1 )θ σ* ( s ) ds, sinh 1 0

(5.12)

0

˜ ( σ ) ) * = σ * – Ᏺ σ ( ξσ* ( x ), ξχ ( nΔ, x ), ξϕ ( ( n + 1 )Δ, x ) + b˜, σ * ), i i i i i i

(Ᏺ

n+1 ( θσ )i

˜ (σ)

Δ ( Ᏺ ) *i + ( θ σ ) i =  , 1+Δ n

1 n+1 σi

∫

= – G ( x i, s )θ σ ( ( n + 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξσ ( ( n + 1 )Δ, x i ) = – cosh ( x i – s )θ σ ( ( n + 1 )Δ, s ) ds – i sinh ( s – 1 )θ σ ( ( n + 1 )Δ, s ) ds; sinh 1 0

0

χ n n+1 n ˜ (χ) n ( Ᏺ ) i = χ i – Ᏺ ( ξσ ( ( n + 1 )Δ, x i ), ξχ ( nΔ, x i ), ξϕ ( ( n + 1 )Δ, x i ) + b˜, σ i , χ i , ρ i ),

( θχ )i + Δ  ( Ᏺ ) i 2 ( θ *χ ) i = , 1+Δ  2 n

˜ (χ)

n

1

∫

χ i* = – G ( x i, s )θ χ* ( s ) ds, 0

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1

∫

∫

cosh x ξχ* ( x i ) = – cosh ( x i – s )θ χ* ( s ) ds – i sinh ( s – 1 )θ χ* ( s ) ds, sinh 1 0

(5.13)

0

χ n+1 ˜ (χ) ( Ᏺ ) *i = χ i* – Ᏺ ( ξσ ( ( n + 1 )Δ, x i ), ξχ* ( x i ), ξϕ ( ( n + 1 )Δ, x i ) + b˜, σ i , χ *i , ρ i ),

n+1 ( θχ )i

˜ (χ)

Δ ( Ᏺ ) *i + ( θ χ ) i =  , 1+Δ n

1

∫

n+1

χi

= – G ( x i, s )θ χ ( ( n + 1 )Δ, s ) ds, 0

xi

1

∫

∫

cosh x ξχ ( ( n + 1 )Δ, x i ) = – cosh ( x i – s )θ χ ( ( n + 1 )Δ, s ) ds – i sinh ( s – 1 )θ χ ( ( n + 1 )Δ, s ) ds. sinh 1 0

0

The algorithm can also be modified for the system of integral equations and for the predictor–correc tor scheme. Computationally, it was found advantageous to replace Eqs. (5.7) and (5.11) with the follow ing relations (see (1.6), (1.7)): 1

∫

n

ϕ i = β G ( x i, s ) [ e

χ ( ( n – 1 )Δ, s )

1 β = , ε

– ρ ( s ) ] ds,

0

xi

∫

(5.14)

1

ξϕ ( nΔ, x i ) = β [ e

χ ( ( n – 1 )Δ, s )

∫

– ρ ( s ) ] ds – β ( 1 – s ) [ e

0

χ ( ( n – 1 )Δ, s )

– ρ ( s ) ] ds,

0

where the Green’s function G(xi, s) is calculated as (see Section 1) ⎧ s ( x i – 1 ), G ( x i, s ) = ⎨ ⎩ x i ( s – 1 ),

0 < s ≤ xi , x i ≤ s < 1.

5.4. Finally, the computational algorithms proposed above for finding an approximate solution of the initial boundary value problem (3.1)–(3.5) were verified using orthogonal sweep method. The preliminary steps preceding the use of sweep method are described as applied to model problem (5.1). By making the substitutions U ( x ) = Φ,

V ( x ) = ξΦ,

u ( x ) = ϕ,

v ( x ) = ξϕ,

Eq. (5.2) is transformed into the system of two firstorder differential equations ξU ( x ) = V ( x ),

ξV ( x ) = Ꮾ U ( x ) + Ᏺ ,

where Ᏺ = (ξv(x) – u(x) + Δf )/(1 + Δ). This system is rewritten in the matrix form ⎛ ⎞⎛ ⎞ ⎛ ⎞ d⎛  ⎜ U ( x ) ⎟ = ⎜ 0 1 ⎟ ⎜ U ( x ) ⎟ + ⎜ 0 dx ⎝ V ( x ) ⎠ ⎝ Ꮾ 0 ⎠ ⎝ V(x) ⎠ ⎝ Ᏺ

⎞ ⎟. ⎠

(5.15)

For system (5.15), the boundary conditions are specified as ⎛ ⎞ (1 0)⎜ U(x) ⎟ = 0 ⎝ V(x) ⎠

at

x = 0,

⎛ ⎞ (1 0)⎜ U(x) ⎟ = 0 ⎝ V(x) ⎠

at

x = 1.

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ρ(x)

ρ(x)

(b)

(a) 1

1

σ

σ 1 1 1  – Δ1   + Δ1 6 6 6

5 5 5  – Δ1   + Δ1 6 6 6

1

x

11 2 2 1 2  – Δ1   + Δ1  – Δ1   + Δ1 33 3 3 3 3

1

x

Fig. 2. (a) L = 6 × 10–7 m and (b) L = 3 × 10–7 m.

After introducing the notation ⎛ A = ⎜ 0 1 ⎝ Ꮾ0

⎞ ⎟, ⎠

⎛ w = ⎜ U(x) ⎝ V(x)

⎞ ⎟, ⎠

⎛ ⎞ g = ⎜ 0 ⎟, ⎝ Ᏺ⎠

L = R = ( 1 0 ),

boundary value problem (5.15), (5.16) becomes dw/dx = Aw + g, Lw = 0 at x = 0, Rw = 0 at x = 1. (5.17) This problem is solved by applying orthogonal sweep method, which is described in detail, for example, in [19]. The subsequent reasoning concerning the application of sweep method to problem (3.1)–(3.5) is obvious. 6. CONCLUSIONS Several approaches have been described in detail for finding approximate solutions to the ballistic diode problem, which is well known in the physics of semiconductors. The numerical results are discussed in the Appendix. They are feasible from the point of view of physics and are overall consistent with earlier results (see, e.g., [6], [20–25]) and with more recent numerical experiments (see [26]). APPENDIX: NUMERICAL RESULTS Below, we describe the results of the numerical experiments. The formulas for c, c12, …, c22 were taken from [3, 4], while ρ(x) was specified a smoothed piecewise constant function (see Fig. 1). The doping den sity ρ(x) was defined as (see Fig. 2) ⎧ 1, 0 ≤ x ≤ d – Δ 1 , ⎪ ⎪ ρ 1 ( x ), d – Δ 1 ≤ x ≤ d + Δ 1 , ⎪ ρ ( x ) = ⎨ δ, d + Δ 1 ≤ x ≤ 1 – d – Δ 1 , ⎪ ⎪ ρ 2 ( x ), 1 – d – Δ 1 ≤ x ≤ 1 – d + Δ 1 , ⎪ ⎩ 1, 1 – d + Δ 1 ≤ x ≤ 1,

(0.1)

where 15 1 5 2 3 1+δ ρ 1 ( x ) = –  ( 1 – δ )  x˜ –  x˜ + x˜ +  , 16 5 2 3 5 2 3 + δ , ρ 2 ( x ) = 15  ( 1 – δ ) 1 x˜ –  x˜ + x˜ + 1 3 16 5 2

x–d x˜ = , Δ1

x – ( 1 – d, ) x˜ =  Δ1

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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E 2.501 2.251 2.001 1.751

1.501 0 0.16 0.33 0.50 0.68 0.83 1.00 0.17 0.35 0.53 0.70 0.86 1.00 0.09 0.26 0.44 0.61 0.78 0.95 0.08 0.24 0.41 0.59 0.75 0.92 u x x R 1.0 1.349 0.9 1.009 0.8 0.7 0.849 0.6 0.5 0.599 0.4 0.3 0.349 0.2 0.1 0.099 0 0.16 0.33 0.50 0.68 0.83 1.00 0 0.16 0.33 0.50 0.68 0.84 1.00 0.08 0.24 0.41 0.59 0.75 0.92 0.08 0.24 0.41 0.59 0.75 0.92 x x 0

Fig. 3. V = 1 V, L = 6 × 10–7 m, K = 200, ε1 = 10–7, Δ1 = 1/12, and δ = 0.004. ϕ 57.971 52.971 47.971 42.971 37.971 32.971 27.971 22.971 17.971 12.971 7.971 2.971

E 3.231 2.981 2.731 2.481 2.231 1.981 1.731 0

R 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.17 0.35 0.53 0.70 0.86 1.00 0.09 0.26 0.44 0.61 0.78 0.95 x

0 u 1.396

0.16 0.33 0.50 0.68 0.83 1.00 0.08 0.24 0.41 0.59 0.75 0.92 x

1.146 0.896 0.646 0.396 0.146 0

0.16 0.33 0.50 0.68 0.83 1.00 0.08 0.24 0.41 0.59 0.75 0.92 x

0

0.16 0.33 0.50 0.68 0.84 1.00 0.08 0.24 0.41 0.59 0.75 0.92 x

Fig. 4. V = 1.5 V, L = 6 × 10–7 m, K = 200, ε1 = 10–7, Δ1 = 1/12, and δ = 0.004.

and the constants δ, d, and Δ1 are given below. The physical parameters required in the computations were specified as follows: the electron charge qe was 1.6 × 10–19 C; the doping density N+ of the n+ region was 5 × 1023 1/m3; the doping density N of the n region was 2 × 1021 1/m3; the ambient temperature T0 was 300 K; the Boltzmann constant KB was 1.38 × 10–23 J/K; the length L of the n+–n–n+ channel was 6 × 10–7 m or 3 × 10–7 m; the dielectric constant ε was 9.30287 × 10–5 or 3.72115 × 10–4; δ ranged from 1/3 to 0.004; d = 1/6 or 1/3; Δ1 ranged from 1/8 to 1/12. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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DESIGN OF NUMERICAL ALGORITHMS ϕ 77.295 67.295 57.295 47.295 37.295 27.295 17.295 7.295

197

E 11.56 10.56 9.56 8.56 7.56 6.56 5.56 4.56 3.56 2.56 1.56

0.225 0.442 0.658 0.875 0.225 0.433 0.650 0.863 0 0.113 0.333 0.550 0.762 0.988 0.113 0.325 0.542 0.762 0.975 x u x R 1.001 0.24 0.22 0.901 0.20 0.801 0.18 0.701 0.16 0.601 0.14 0.12 0.501 0.10 0.401 0.08 0.225 0.433 0.650 0.863 0.225 0.433 0.650 0.863 0 0 0.113 0.325 0.542 0.762 0.975 0.113 0.325 0.542 0.762 0.975 x x 0

Fig. 5. V = 2 V, L = 3 × 10–7m, K = 100, ε1 = 10–6, Δ1 = 1/12, and δ = 0.004.

ϕ 38.647 33.647 28.647 23.647 18.647 13.647 8.647 3.647

E 2.761 2.511 2.261 2.011 1.761 0

R 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0

0.17 0.34 0.51 0.68 0.84 0.09 0.26 0.42 0.59 0.76 0.93 x

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

1.511 u 0.201 0.181 0.161 0.141 0.121 0.101 0.081 0.061 0.041

0

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

0

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

Fig. 6. V = 1 V, L = 6 × 10–7 m, K = 200, ε1 = 10–6, Δ1 = 1/12, and δ = 1/5.

The dimensionless variable b˜ (bias across the diode) was given by Vq e b˜ =  . KB T0 Here, V (Volt) is the dimensional bias across the diode (see Fig. 1) and the constants qe, KB, and T0 are given above. An approximate solution to systems (3.1)–(3.5) and (4.1)–(4.4) was found by applying the algorithms described in Section 5. For this purpose, these algorithms were implemented as software codes developed in Object Pascal in the Delphi 6 environment. The codes include the following operations. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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BLOKHIN et al. ϕ 38.647 33.647 28.647 23.647 18.647 13.647 8.647 3.647

E 4.658 3.658 2.658 1.658 0

R 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

0

0.17 0.33 0.49 0.66 0.82 0.99 0.09 0.25 0.41 0.58 0.74 0.90 x

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

0 u 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

0

0.17 0.33 0.49 0.65 0.81 0.97 0.09 0.24 0.41 0.57 0.73 0.89 x

Fig. 7. V = 1 V, L = 3 × 10–7 m, K = 1000, ε1 = 10–6, Δ1 = 1/12, and δ = 1/3.

1. Initialization: (a) the parameters characterizing the computational algorithm: K is the number of grid nodes; Δ is the mesh size in t; ε1 is the accuracy; i.e., the algorithm halts if K

∑

K

n+1

ϕi

n

– ϕi ≤ ε1 ,

i=1

∑

K

n+1

σi

n

– σi ≤ ε1 ,

i=1

∑χ

n+1 i

n

– χi ≤ ε1 ,

i=1

n

where ϕ i = ϕ(nΔ, xi), etc., xi = ih, i = 1, K – 1 , Kh = 1; (b) the parameters characterizing the original problem: ( b˜ ); (c) initial data for the unknown functions. 2. Storage of the physical parameters necessary in the computations. System (4.1)–(4.4) was computed on a grid with the variable step hk = xk + 1 – xk, k = 0, K – 1 (see Section 5). The interval [0, 1] was divided into five subintervals [0, d – Δ1], [d – Δ1, d + Δ1], [d + Δ1, 1 – d – Δ1], [1 – d – Δ1, 1 – d + Δ1], and [1 – d + Δ1, 1], each of which had K/5 nodes and a uniform grid. The integrals of the form F(x) =

∫

x 0

xi

Fi =

∫ 0

f ( s ) ds were evaluated using the trapezoidal rule i–1

f ( s ) ds = h

∑

j=0

fj + fj + 1 h   =  f0 + h 2 2

i–1

h

∑ f + 2 f j

j=1

i

fi – 1 + fi = F i – 1 + h  , 2

where Fi = F(xi) and fi = f(xi). The computations were performed with different parameters by applying different algorithms. The most difficult task was to find approximate solutions at δ = 0.004. The computational algorithm developed for system (4.1)–(4.4) performed this for L = 6 × 10–7 m, while its modification, for L = 3 × 10–7 m. The approximate solutions obtained for the electric potential and the electron energy, density, and velocity are shown in Figs. 3–5. The other approaches (based on spline functions, the predictor–corrector schemes, and orthogonal sweep method) produced good results for other values of δ (from 1/3 to 0.05). It was found that the spline function algorithm converges faster than the orthogonal sweep method, which, in turn, is faster than the predictor–corrector scheme. Since the behavior of the plots is similar and, in principle, the orthogonal COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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sweep method yields results with guaranteed accuracy, we present the numerical results produced by this method (see Figs. 6, 7).

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 070100585), by the Siberian Branch of the Russian Academy of Sciences (interdisciplinary integrated project no. 91, the year 2009), and by the program “Development of the Scientific Potentials of Higher Educational Institu tions” the 2009–2010 years (project no. 2.1.1/4591).

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21. A. M. Anile, M. Junk, V. Romano, and G. Russo, “CrossValidation of Numerical Schemes for Extended Hydrodynamical Models of Semiconductors,” Math. Models Methods Appl. Sci. 10, 833– 861 (2000). 22. A. M. Anile, V. Romano, and G. Russo, “Extended Hydrodynamical Model of Carrier Transport in Semicon ductors,” SIAM J. Appl. Math. 61, 74–101 (2000). 23. C. L. Gardner, “Numerical Simulation of a SteadyState Electron Shock Wave in a Submicrometer Semicon ductor Device,” IEEE Trans. Electron Devices 38, 392–398 (1991). 24. C. L. Gardner, J. W. Jerome, and D. J. Rose, “Numerical Methods for the Hydrodynamic Device Model: Sub sonic Flow,” IEEE Trans. ComputerAided Design 8, 501–507 (1989). 25. V. Romano and G. Russo, “Numerical Solution for Hydrodymamical Models of Semiconductors,” Math. Models Methods Appl. Sci. 10, 1099–1120 (2000). 26. V. Romano, “2D Numerical Simulation of the MEP EnergyTransport Model with a Finite Difference Scheme,” J. Comput. Phys. 221, 439–468 (2007).

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