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On a family of finite-difference schemes with discrete transparent boundary conditions for a parabolic equation on the half-axis

An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averagings both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.


Introduction
In many applications, a problem of solving partial differential equations in unbounded domains arises. A number of approaches to the problem is developed mainly associated with the statement of additional boundary conditions on artificial boundaries [1]- [6]. The conditions are called the (exact) artificial/non-reflecting/transparent boundary conditions provided that they are satisfied by the solutions of the original problems in unbounded domains. For definiteness, we exploit the last name (TBCs). For parabolic evolution equations or for the Schroödinger equation, the TBCs are integro-differential relations along the artificial boundaries. Their adequate discretization is nontrivial since it can produce significant reflections from the boundaries and even instability in computations as well as create difficulties in rigorous proofs of stability of the resulting numerical method.
An alternative approach suggests to implement the idea of the TBCs on the mesh level. Namely, first one consider a discretization of the problem on an infinite mesh in the unbounded domain (which is not practical because of the infinite number of unknowns). Its solution is restricted to the finite mesh by deriving a mesh counterpart of the TBC on the artificial boundary. One version of this approach is associated with the derivation of discrete TBCs requiring to solve analytically model mesh problems on infinite grids.
This approach had worked well by the complete absence of reflections from artificial boundaries and reliable stability of computations in practice as well as by clarity of the mathematical background and rigorous proofs of stability of the resulting mesh method in theory. Such an approach is developed in detail for 1D time-dependent Schrödinger equation in [7]- [11] and also used for 1D parabolic equations in [1,12,8,6,13].
In this paper, the approach is developed for the 1D self-adjoint parabolic equation on the half-axis. We study a broad family of two-level finitedifference σ-schemes with averaging in space with a weight θ. We prove its stability by the energy method and rigorously derive the discrete TBC by applying the method of reproducing functions. Notice that both points are not so well developed in some other papers. Similarly to [11], this allows one to cover in a unified manner a collection of particular schemes: the standard scheme without averaging (θ = 0), the linear finite-element method (θ = 1 6 ), scheme of higher order of accuracy (for constant coefficients, θ = 1 12 ) and a vector scheme on a four-point stencil (θ = 1 4 ). The results generalize those obtained for θ = 0 in [13] but here we prove stability in two (not one) energy norms, the rigorous derivation of the discrete TBC is notably different and results of numerical experiments are included as well.
Notice that the results for the family of finite-difference schemes can be enlarged for the 2D (or multi-D) case exploiting the technique from [14,15].
2 An initial-boundary value problem and a finitedifference σ-scheme with averaging in space and an approximate TBC We consider the one-dimensional parabolic equation for x > 0 and t > 0. Its coefficients satisfy the conditions ρ(x) Equation (1) is supplemented with the following boundary condition, the condition at infinity and the initial condition: We assume that the coefficients become constants as well as f and u 0 vanish for sufficiently large x X 0 (for some X 0 > 0) An integro-differential TBC satisfied by the solution to this problem can be written in the Dirichlet-to-Neumann form for t > 0 and for any X X 0 ; other equivalent forms are also known. The TBC is nonlocal in time; recall that the involved operator serve as an (abstract) approximate TBC for (5) at the node x J ; here we have discretized ∂u ∂x with weight in t and symmetrically in x. We first write down equation (6) on the mesh ω h ∪ {x J } and apply it at the node x J only in order to eliminate the values U (σ) J+1 involved in the left-hand side of (10). Namely, since (4) we get the boundary condition (8). Importantly, the boundary condition (8) for S = 0 is the natural approximation of the Neumann boundary condition for this finitedifference scheme for x = X. Such an approach was implemented earlier in [9,13,11]; it reliably leads to the computationally stable form of discrete TBCs (in contrast to some other approaches).
The corresponding three-point system of mesh equations for a vector {U m j } J j=0 of the solution values on the upper level, has the form: compare with [11]. Here the coefficients are given by formulas for 1 j J; in particular, for j = J, these expressions become more simple Equation (11) is written assuming that the operator in the approximate TBC 3 Stability of the finite-difference scheme on the finite mesh with the approximate TBC We consider the stability problem for the finite-difference scheme (6)- (9) with respect to the initial data U 0 h , the free term F and a perturbation in the boundary condition (8) and take g(t) = 0. We need to introduce several mesh counterparts of the L 2 (Ω)-inner products holds for any M 1 and for any decomposition The bound holds also in the case θ = 1 4 provided that F (0) = 0 (one has to drop the summand with F (0) ).
Consequently, for σ 1 2 and θ 1 4 , the scheme has a unique solution.
Proof. We take the (· , ·) ω h -inner product of equation (6) and a function W ∈ H 0 (ω h ), sum the result by parts (using the second assumption (4)) and Choosing W = U (σ) m and applying the boundary condition (13) and other assumptions (4), we get We multiply the result by τ m and sum up it over m = 1, . . . , M. Applying the formula U (σ) = U (1/2) + (σ − 1 2 )τ ∂ t U , we obtain the first energy equality for M 1, where , we sum the result by parts and derive the following bound Using the left-hand inequality (12), conditions σ 1 2 and (14) and applying the standard argument lead from the first energy equality (17) to bound (15). It is well-known that such a bound implies the existence and uniqueness of a solution to the finite-different scheme.
We also define a symmetric bilinear form Proposition 2 Let U be a solution to the finite-difference scheme (6)-(9) with the generalized boundary condition (13) instead of (8). Let the operator S satisfy an inequality for any function Φ given on ω τ such that Φ 0 = 0. Then, for σ 1 2 and θ < 1 4 , the second energy bound holds for any M 1 and any decomposition The bound holds also in the case θ = 1 4 provided that F (0) = 0. Consequently, for σ 1 2 and θ 1 4 , the scheme has a unique solution.
Proof. We choose W = ∂ t U m in (16), apply the boundary condition (13) and assumptions (4) and get We multiply the equality by τ m and sum up it over m = 1, . . . , M. Applying again the formula for M 1, where , we sum the result by parts in t and x and get Using the left-hand inequality (12), conditions σ 4 Stability of the finite-difference scheme on an infinite mesh In order to construct and study the discrete TBC, we first turn to the finite-difference scheme on an infinite mesh for the original problem (1)-(3) on the half-axis Assumptions (4) are supposed to be fulfilled. Let g(t) = 0 and U 0 h | j=0 = 0. We introduce the Hilbert spaces H h and H h (mesh counterparts of L 2 (R + )) consisting of functions W given on the meshes respectively ω h,∞ (and with W 0 = 0) and ω h,∞ and such that W 2 We define symmetric bilinear forms [11] holds for all W ∈ H h and θ 1 4 [11]. We also define the mesh counterparts of the norm in L 2 (R + × (0, t M )) and, for σ 1 2 and θ < 1 4 , the first energy bound holds for any M 1. Here The bound holds also in the case θ = 1 4 provided that F (0) = 0.
Proof. We extend A h and C θ [ρ h ] up to operators acting in H h by setting (A h W ) 0 := 0 and (C θ [ρ h ]W ) 0 := 0. By virtue of assumptions (4) and the for any W, V ∈ H h . To establish equality (25), one can first transform the finite sum  (16)) and then pass to the limit as j 1 → ∞ using the property lim j→∞ W j = 0 for W ∈ H h (as in [13] for θ = 0). Now we rewrite equation (21), together with the homogeneous boundary condition (22), as an operator Thus compare with (17). Similarly to the proof of Proposition 1, it implies bound (24).
We define the mesh counterparts of the norm in L 2 (X, ∞) such that concerning the correctness of the last definition, see [11].
holds for all M 1. Its right-hand side is nonnegative for σ 1 2 .
Proof. By virtue of equation (21) at the node x J with F m J = 0, relation (10) is equivalent to the boundary condition (8); thus, the solution to the scheme (21)-(23) satisfies the scheme (6)-(9) as well. Taking the difference of the energy equalities (28) and (17) (with G = 0) and applying simple identities for any W ∈ H h and κ h = ρ h , c h , we obtain the announced equality.
We also derive stability in another norm.
holds for the solution U m ∈ H h , for all m 0, to the finite-difference scheme (21)-(23) and any M 1. Here The bound holds also in the case θ = 1 4 provided that F (0) = 0.

Proof. The second energy equality
holds, compare with (20). Similarly to the proof of Proposition 2, it implies bound (30).
Corollary 2 Under the hypotheses of Corollary 1, an equality holds for any M 1. Its right-hand side is nonnegative for σ 1 2 .
Proof. The result is derived by taking the difference of (31) and (20) (with G = 0). By definition, the discrete TBC is an approximate TBC (10) with the operator S = S ref . It will be explicitly constructed in the next section. Corollaries 1 and 2 clarify the energy meaning of conditions (14) and (18) for the discrete TBC, for Φ = U J , and are exploited below to prove the conditions (for any Φ).

Derivation and analysis of the discrete TBC
Now we turn to derivation of the explicit form for the discrete TBC in the form (10) and verification of inequalities (14) and (18) for it. We confine ourselves by the case of the uniform mesh ω τ , i.e., τ m = τ for m 1. Consider an auxiliary finite-difference problem on the uniform part of the infinite mesh in x for some q 0 1. Here the limiting finite-difference operator has appeared. We seek for the solution satisfying the following property for sufficently large q > q 0 . Since the coefficients are constant and the meshes are uniform, the stated problem can be solved explicitly. For a mesh function Φ: ω τ → C such that |Φ| ∞, q < ∞ for some q > 0, recall the reproducing function i.e., analytic in the disc D 1/q := {|z| < 1 q } ⊂ C, satisfying a bound Conversely, for a function p ∈ A(D r ) (for some r > 0), the transformation dϕ for any m 0, 0 < r 1 < r, (37) is well defined implying the Cauchy inequality Hereafter i is the imaginary unit, and Re z and Im z are real and imaginary parts of z ∈ C.
Taking into account conditions (34) and (35), for |z| < r = 1 q , we calculate The coefficients γ(z) and d(z) are expressed by formulas By virtue of (32) and (39) a difference equation holds. The corresponding characteristic equation has the form By virtue of (38) and (36) we get that, for any j J and m 1, bounds Therefore bound (43) holds with C = C 1 1−q 0 r 1 , where For real Φ, the functions Φ(z) and γ(z) are real as well for z ∈ R. If in addition |z| r, then |γ(z)| > 1 and thus ν 1 (z) = Z −1 1 (γ(z)) is real. Therefore U is real too, see (37).
The proofs of Lemma 1 and Proposition 5 remain valid also for σ = 0, θ < 1 4 and θ = 0. We go back to the derivation of the discrete TBC. By virtue of formula (44) we have Therefore it is easy to check that a formula holds, compare with (39). By virtue of the well known formula for the multipklication of two poer series it leads to the discrete TBC (10) with the operator S = S ref of the discrete convolution form with the kernel Let us see that Propositions 1 and 2 on stability are valid for S = S ref .
Proof. We apply a method first suggested in [9]. Fix any M 1 and real values Φ 1 , . . . , Φ M . Extend Φ m = 0 for m = −1, 0 and m > M. We define a function U by formula (42) for j J − 1 and set, for example, U m j := 0 for 0 j < J − 1 and m 0, and then set F : By virtue of Proposition 5, the constructed function U serves as the real solution to the problem (32)-(34) and therefore as one to the scheme (21)-(23), where F = 0 on (ω h,∞ \ ω h ) × ω τ and U 0 h = 0. Then Corollary 1 implies inequality (14) whereas Corollary 2 implies inequality (18) since U m J = Φ m for 0 m M.
The case σ = σ 0 1 2 will be also covered in Remark 4 below. We find the kernel R explicitly. We introduce quantities by recurrence formulas p 0,α,β = 1, p m,α,β = 0 for m < 0 (49) with parameters α, β. Their close connecton to the classical Legendre polynomials will be shown below.
Proposition 7 For σ > 0, σ = σ 0 or σ = σ 0 1 2 , and θ 1 4 , the kernel R of the operator of the discrete TBC (46) is real and is expressed by an explicit formula with the quantities α, β and δ of the form Proof. Let first σ = σ 0 . By Lemma 1 for z ∈ D r we have One can straightforwardly verify that where the coefficients d 0 , d 1 , α 0 , α 1 and α, β, δ are given by formulas (51)-(54). Let first also α = 0. We write down a formula where κ = √ α and µ = β √ α are real for α > 0, or κ = i |α| and µ = −i β √ |α| are purely imaginary for α < 0; herewith κ 2 = α and κµ = β are always real. Inserting the formula into (56) leads to at least for sufficiently small z (in accordance with the proof of Lemma 1), where + √ w is an analytic branch of √ w on C with the cross-cut along the negative real half-axis Re w < 0 such that + √ 1 = 1. Formulas (55) and (57) imply an equality The following generalized formula for the reproducing function of the Legandre polynomials holds [9]: for any κ ∈ C, integer l 0 and sufficiently small z, that easily follows from the classical one in the case κ = 1, l = 0 [16]. Therefore where P m (µ) ≡ 0 for m < 0. Applying the recurrence relation for the Legendre polynomials one can simplify the last formula as follows i.e., to derive a formula Following [6,13], we introduce modified Legendre polynomials p m,κ (z) := κ m P m (z). From (58) clearly p m,α,β = p m,κ (µ) satisfy recurrence equalities (48) and (49) and, in particular, they are real. Therefore formula (50) is proved. For α = 0 formula (57) is simplified and takes the form Hence one can easily verify that formulas (48), (49) and (50) remain valid and even are simplified in the case α = 0.
Owing to continuous dependence of U on σ 1 2 (see Remark 3) and R on σ 1 2 (it is clear), one can pass to the limit as σ → σ 0 on the left-hand side of the discrete TBC (10) with S = S ref and on the right-hand side of equality (46). This justifies the validity of formula (46), for R of the form (50), in the case σ = σ 0 1 2 (that is possible only provided that Remark 4 Proposition 6 remains valid also in the case σ = σ 0 1 2 . To see this, it suffices to insert formula (46) into inequalities (14) and (18) for S = S ref that have been already proved, for σ = σ 0 , and pass to the limit as σ → σ 0 taking into account the continuous dependence of R on σ.
In practical computations, recurrence equalities for R are more convenient than formula (50).

Proposition 8
The kernel R satisfies the recurrence equalities Proof. The form of formula (59) differs from the corresponding one in [11], Proposition 5.7 only by a constant multiplier. Therefore Proposition 5.8 in [11] implies a recurrence formula Inserting κµ = β and κ 2 = α leads to (60). Formulas (61) straightforwardly follow from (50)   One can rather easily extend the above results are to the case of the third boundary condition at x = 0, or to the Cauchy problem where equation (1) is posed on R. The case σ < 1/2 could be also analyzed under suitable additional condition between steps τ and h j .
Note that |u 0 (x)| < 3.8 · 10 −6 for x X. On Fig. 2, we demonstrate the numerical solution and its error computed for σ = 1 2 , θ = 1 12 , h = 0.05 and τ = 1 1500 .  Notice that if one sets simply the Neumann boundary condition (i.e., takes S = 0) instead of the discrete TBC at x = X, then it is necessary to increase X three times to reach the error of the same order of smallness (for the same h and τ ); herewith the maximum absolute error is reached at (x, t) = (X, T ) (the corresponding graphs are omitted).
In Example 2, we take the data g(t) = t 2 and u 0 (x) = 0. The exact solution to such problem u 2 (x, t) = 32t 2 I 4 (ξ) is also known and is calculated by applying the recurrence formulas [17] I 0 (ξ) = erfc(ξ) ≡ 2 √ π ∞ ξ e −ζ 2 dζ, I 1 (ξ) = 1 √ π e −ξ 2 − ξ erfc(ξ), I n (ξ) = 1 2n I n−2 (ξ) − ξ n I n−1 (ξ), n = 2, 3, 4, where ξ = x 2 √ t . Choose T = 1 and X = 1. On Fig. 3 we present the numerical solution and its error computed for σ = 1 2 , θ = 1 12 , h = 0.1 and τ = 0.01. In contrast to Example 1, now the solution is not close to 0 for x = X. The error is maximal at the node (x j , t m ) = (h, τ ) (but not on the artificial boundary x = X). Notice that decreasing of X down to 0.2 for the same mesh steps does not increase the error (for u 0 = 0 and while applying namely the discrete TBC, this is natural and clear from above). Moreover, if once again one sets simply the Neumann boundary condition instead of the discrete TBC at x = X, then (for the same h and τ ) the absolute error equals 0.15 and is unacceptably large. It decreases to the same values as on Fig. 3 only when X increases five times. On Fig. 4 we give the graphs of errors in the cases θ = 0 and θ = 1 6 . Their forms are different and the maximum absolute errors are about two orders of magnitude greater than in the case θ = 1 12 . In addition in Tables 2 and 3 we put the absolute errors in dependence with M for h = 0.1 and various θ. For θ = 0, 1 6 and 1 4 the errors do not practically change already for M 50; herewith their minimum values for θ = 0 and θ = 1 6 are close whereas for θ = 1 4 one is approximately twice larger.   2.638 · 10 −7 6.797 · 10 −8 4.032 · 10 −8 8.962 · 10 −8 1.062 · 10 −7