Exact diﬀerence schemes and schemes of higher order of approximation for convection-diﬀusion equation. I

– The initial-boundary value problem for a convection-diﬀusion equation


Introduction
In constructing a difference scheme the main aim is to approximate an original problem with a prescribed accuracy in a finite number of operations. In this regard, the question of the approximation order of the difference scheme arises at once. The order of the approximation is desired to be as high as possible with a minimum number of grid nodes in the pattern of a scheme at the same time. In some cases, the EDS can be constructed.

Definition 1.
A difference scheme is exact if the truncation error equals zero or y = u at the grid nodes.
Let C m n (Q T ) be the class of functions with m derivatives in x and n derivatives in t, all necessary derivatives being continuous in the domain Q T = Q T ∪ ∂Q T .
Contracting EDSs was, for example, discussed in [2,3,4,5,6]. In the papers by R.E. Mickens certain rules for construction of the nonstandard finite difference schemes are given. In [2] there were constructed EDSs from which the implementable truncated difference schemes were derived. For the Cauchy problem the following EDS can be constructed [7] y n+1 − y n τ In the above scheme the special Steklov averaging is used.
The authors have established that for the parabolic problems with travelling wave solutions the EDS may be constructed [7,8]. This paper refers to the travelling waves which arise in many problems such as heat transfer, combustion, reaction chemistry, fluid dynamics, plasma physics, soil-moisture, foam drainage, crystal growth, biological population genetics, cellular ecology, neurology and synergy [9,10]. This was the motivation for constructing EDS for a convection-diffusion problem. The authors took advantage of the equivalence of the convection-diffusion equation to the transport equation for travelling wave solutions and constructed the difference scheme which can be reduced to the EDS for transport problem in this special case. The main goal was to construct the difference scheme which can be applied not only for a class of travelling wave solutions but also for a wide class of solutions. The monotonicity of the scheme was also an important question. For further investigation the corollary of the maximum principle was used, which asserts that for a solution to the difference equation the estimate  [11]. The equation (1) can be rewritten in a more general unified canonical form [11] A(P )y(P ) = Q∈P att (P ) where A(P ), B(P, Q), F(P ) are given grid functions, P att (P ) = P att(P ) \ {P }, and P att is the difference pattern.
[11] The difference scheme (3) is monotone if A(P ) > 0, B(P, Q) > 0 for all P ∈ ω, Q ∈ P att (P ) and The outline of this paper is as follows. In the first two sections the EDS for transport problems are considered. On their basis the EDS for a convection-diffusion problem is constructed in the next section and the approximation order is considered. The numerical results are given and the future research directions are discussed.
2 Difference schemes for a homogenous transport equation T ]} let us consider the initialboundary value problem for a transport equation with a positive coefficient a > 0 where Q T = {(x, t) : x ∈ (0, l], t ∈ (0, T ]}. It is well-known that the solution of the problem (4) is a travelling wave where a is the wave velocity and U (ξ) is the differentiable function [11]. In this section we review some of the difference schemes approximating the problem (4) -(5) with particular reference to an exactness property. The difference schemes introduced below are exact under the same condition connected with the Courant number γ γ = aτ /h = 1.
Let us begin with the well-studied explicit difference scheme [11,12] y t + ay x = 0, This scheme is monotone and stable for γ ≤ 1. Another scheme [11,12] y t + aŷ x = 0, is monotone and stable with a contrary condition ie. γ = aτ /h ≥ 1. The above schemes have the first order of approximation ψ n i = O(τ + h) for γ = 1. The third scheme approximating problem (4) -(5) is weighted simultaneously with respect to the time and space and has the form Here y (α) = αy n i + (1 − α)y n i−1 . This scheme is also exact under the condition γ = 1, but only for a special case of weights, ie. α + σ = 1. It has the second order of approximation O(τ 2 + h 2 ) for α = σ = 0.5 and γ = 1. Furthermore, it is monotone and stable under the condition α + γσ ≥ max {γ, 1} [13]. Now we deal with another difference scheme weighted simultaneously with respect to the time and space This scheme is a combination of the schemes (6) and (8). It is monotone and stable for [8] This scheme is particularly important in view of next sections. Because of it, we will recall the following lemmas. Lemma 1. If γ = 1, then the difference scheme (10), (7) is exact.
Lemma 2. The approximation error of the difference scheme (10), (7) is Proof. Lemma 1 shows that ψ n i = 0 for γ = 1. Now, let γ = 1. The approximation error fulfills the relation Our first goal is to estimate the error ψ n 1i . The difference operator u t has a Taylor expansion of the form u t,i = ∂u ∂t + O(τ 2 ), so the approximation error ψ n 1i is Next, let us estimate the error ψ n 2i . In view of the Taylor expansionû Analogously, we can show that The trick of the proof is to estimate the expression ψ n The only, but crucial use of the above equality is that from (11) -(13) we obtain the estimations It is straightforward to show that the error of approximation is . Thus, the difference scheme (10), (7) has the second order of approximation for σ = 0.5.

Difference schemes for a semilinear transport equation
In the domain Q T consider a semilinear transport equation with the initial and boundary conditions (5). The difference scheme approximating the above problem is [8] y t,i + aσy n+1 where It is exact for γ = 1 [7,8].
If the integrals in the EDS cannot be evaluated exactly, to approximate them the Euler-Maclaurin formula can be applied [14]. In this case, instead of EDS, we obtain a difference scheme of an arbitrary order of accuracy.
Example. We consider the function u(x, t) = sin 2 (π(x − t))/(1 − t sin 2 (π(x − t))) (see Fig. 1) that satisfies the problem (17), (5) [15]. The corresponding difference scheme is  Table 1 presents the numerical results for time and space steps satisfying the condition γ = 1 and demonstrates the exactness of the difference scheme.  Table 2 presents the numerical results for the different time and space steps for which γ ≤ 1 and demonstrates the second order convergence of the difference scheme.

Exact difference scheme for a convection-diffusion in a one-dimensional problem
In this section the convection-diffusion equation is considered and the difference scheme is constructed. This scheme can be applied for a wide class of solutions but  it is particularly important for travelling wave solutions. In this case the constructed scheme is exact.
In the domain Q T let us consider the boundary-value problem for the convectiondiffusion equation where On the uniform grid ω hτ the problem (23), (21), (22) is approximated by the difference scheme with the weight where the operators Λ, Λ − are given by the formulas It is worth to noticing here that the explicit scheme (σ = 0) is also exact but unstable. The experiments show that, when there is no restriction on the grid steps, the weighted scheme (24) -(26) is unstable for σ < 0.5. Let us denote the error of the method by z = y − u. Then the difference problem for this error takes the following form i , u n θ4,i = u n i + θ n 4i z n i and the approximation error is Here and after θ n pi = const, 0 < θ n pi < 1. The considered scheme is monotone and stable in a linear approximation for τ ≤ h 2 /(ah + 2b(1 − σ)) and 0 ≤ σ ≤ 1. Moreover, it is also stable for σ ≥ 1 and aστ /h ≤ 1. In this case the linear approximation is treated as a finite difference scheme y t + ay x = by

Approximation of the scheme (24) -(26)
First we prove the theorem on an order of approximation of the difference scheme (24) -(26).

Theorem 1. The approximation error of the difference scheme (24) -(26) is O(h + τ ).
Proof. The approximation error of the difference scheme (24) -(26) satisfies the relation It was shown in Lemma 2 that to the error ψ n Taking into account the following estimation and transformations we are now in a position to estimate the error ψ n ). An argument similar to the one used above shows that . We use the relation (14) to obtain the estimations ). Thus, the difference scheme (24) -(26) has the first order of approximation.

Remark 1.
A difference scheme of the second order O(τ 2 + h 2 ) of approximation for α = σ = 0.5 has the form For α = 0 the above scheme reduces to the scheme (24). Scheme (27) where a, b are the positive constants, which is monotone and stable under two conditions: (1 − α)). The first condition is connected with the so-called Peclet number P e = ah/b and for a wide class of problems imposes very small grid steps. As a simple example, the convection-diffusion problem with the small parameter, i.e., b 1 can be considered. Another way to increase the order of approximation and avoid the restriction on the cell Peclet number at the same time is constructing a difference scheme with the regularization where . The approximation error of the above scheme is O(τ + h 2 ) for σ = 0.5. For k(u) ≡ b, the scheme (28) reduces to More information about the difference schemes with a regularization can be found in [11].

The proof of the main result
In the present section the exactness of the scheme (24) -(26) will be considered. First we will need the following lemma.
Proof. First, let us notice that for the travelling wave solution u( From the above equality we get Integrating the equation (29) for t = t n+1 on the closed intervals After some tedious manipulation we obtain In a similar way we get The essential observation is that the above Lemma gives the condition on equivalence of the convection-diffusion equation and the transport equation. In the next theorem we give the condition under which the scheme (24) -(26) is exact. Proof. From Lemma 3 the following equality follows , and the approximation error satisfies the relationships  y i , i = 1, . . . , N −1. The initial approximation is taken from the explicit scheme (24) (σ = 0).

Numerical experiment for a linear equation
In this section we carry out numerical experiments to illustrate the convergence of the exact difference scheme for a linear convection-diffusion equation.
Consider the equation (20) with k(u) = b and b = const > 0 together with the input conditions The solution of this problem is u(x, t) = e 0.5((a+0.5b)t−x) . In this case c = 0.5b and the considered scheme is exact for (a + 0.5b)τ /h = 1. Moreover, for σ = (2a + b)/(2b) and (a + 0.5b)τ /h = 1 the scheme has the second order of approximation.  The number of iterations needed in the nonlinear scheme is sufficiently small. In this section we carry out numerical experiments to illustrate the convergence of the exact difference scheme for a nonlinear convection-diffusion equation.
Consider the equation (20)  In this case c = D and the considered scheme is exact for (a + D)τ /h = 1.  In the future our research will focus on the two-dimensional case of the convectiondiffusion equation. The travelling wave solution will be still considered and the equivalence to the transport equation will be crucial.