Difference schemes of arbitrary order of accuracy for semilinear parabolic equations

– The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u ( x,t ) = X ( x ) T 1 ( t ) + T 2 ( t ) , ∂u∂x (cid:2) = 0 , it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper.

ordinary differential equations, respectively. It is worth mentioning here the paper by Mickens [9] in which nonstandard finite difference schemes are introduced. In [10] the investigations of the order of approximation, stability, and convergence of the high accuracy difference schemes for the nonlinear transfer equation ∂u ∂t + u ∂u ∂x = f (u) have been made. The EDS and the difference schemes of an arbitrary order of approximation for the parabolic equations with travelling wave solutions u(x, t) = U (x − at) were constructed in [7,8].
It was earlier established that for the problems in the parabolic equations with separated variables solutions u(x, t) = X(x)T 1 (t) + T 2 (t) the EDS and the ADSs may be constructed [3,7]. The main aim of this paper is to investigate approximation, stability and convergence of the nonlinear scheme of an arbitrary order of accuracy.
From (3) we find the curve x = x(t), along which we get from (4) the solution u(x, t) = u(x(t), t) of the problem (1)- (2). Here x(0) = x 0 ∈ R is the initial state of the curve x = x(t).
The ADS applying the trapezoidal rule is proposed in the case when the integrals in the EDS cannot be evaluated exactly. We prove that the orders of approximation and accuracy depend only on time step τ divided by the natural constant m ≥ 1. We also consider the convergence of the iteration method which is used to find the solution of the nonlinear scheme. At the end we present the results of the numerical experiments which illustrate theoretical results stated in the paper. We also construct a difference scheme of an arbitrary order of accuracy applying the Euler-Maclaurin formula instead of the trapezoidal rule [2]. We investigate numerically that the error of this method equals O Let us assume that problem (5) -(8) has a unique solution u ∈ C 4 1 Q T with all derivatives bounded and (7) is the key relation. Taking into account these assumptions, we rewrite problem (5)-(6) in the following form We assume that

The difference scheme of an arbitrary order of accuracy
In this Section, the EDS approximating problem (5) -(8) is considered. The ADS is constructed in the case when the integrals in the EDS cannot be evaluated exactly. The trapezoidal rule is applied to approximate the integrals.
Let us introduce space and time grids In [3,7] the EDS approximating problem (5) - (8) was constructed Only in some cases the integrals in the EDS can be evaluated exactly. In other cases, applying the trapezoidal rule, we approximate problem (9) -(10) by the difference scheme where m ≥ 1 is a natural number and ϕ 1 , ϕ 2 , ϕ 3 : R 2 → R are introduced as follows Let us denote the approximate value of the curve x i (t) on n − th level by x n hi , and the exact value of this curve on n − th level by x n i . The curve x i (t) is the solution of problem (9) with the initial value x 0 i . Here y n i is the approximate solution of problem (10) in the node (x n hi , t n ) and u n i , u n hi are the values of exact solution of this problem in the nodes (x n i , t n ) and (x n hi , t n ), respectively.
Equation (13) introduces the moving grid in the domain Q T Let us denote the error of the method by Then the difference problem for the error of the method is written in the following form Here and after θ n pi = const, 0 < θ n pi < 1. The approximation error of the difference scheme (13) -(14) equals

Approximation
In this Section, we prove the theorem on an order of approximation of difference scheme (13) - (14).
Since (16), the function 1 c(x) has a constant sign for all x ∈ R. Thus, the following estimate is valid

Some tedious manipulations yields
Thus, difference equation (13) approximates differential problem (9) with the second order with respect to τ Analogously, we can show that difference equation (14) approximates differential problem (10) with the second order with respect to τ m max 0≤n<N0 Finally, we find that difference scheme (13) -(14) approximates differential problem (9) -(10) with the second order with respect to τ

Stability
In this Section, stability of the ADS is investigated. Let us perturb the initial data of problem (13) -(14) We give a theorem on stability of difference scheme (13)-(14).

Convergence
In this Section, we prove the theorem on an order of accuracy of difference scheme (13) - (14).
Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 06/04/2022 10:22:55 U M C S Theorem 4. Let the assumptions of Theorems 2 and 3 holds. Then for sufficiently small τ ≤ min {τ * 0 , τ * * 0 } the solution of the difference scheme (13) - (14) converges to the solution of the differential problem (9) -(10) and the estimates hold PROOF. Similarly to that before, the functions ϕ 1 , ϕ 3 ∈ C 1 (R 2 ) have bounded derivatives where M p are positive constants. From the equation for the error of the method the following estimates hold The above inequalities are valid for any i ∈ Z, thus To obtain the estimation of the error δu n hi a little manipulation is needed where δy n i = y n i − u n i satisfies the equation Our problem reduces to estimating the error δy n i . A reasoning similar to that used in the first part of the proof, shows that We get Finally, estimates (27) and (28) express the convergence of the solution of difference scheme (13) - (14) to the solution of differential problem (9) -(10) with the second order with respect to τ m .

Iteration method
In this Section, convergence of the iteration method for difference scheme (13) -(14) is discussed.
The following iteration method is used in connection with solving nonlinear difference scheme (13) -(14)  PROOF. Let τ be chosen sufficiently small First, we estimate the difference between 1 x n+1 hi and the initial approximation Now, let us estimate the difference between two successive iterations We are now in a position to estimate the The above inequality is valid for any i ∈ Z, so we get It remains to show that The proof is similar to that used above. Thereby the iterations converge with the rate of geometric progression and the limits exist lim

Numerical examples
In this Section, the example of the previously considered ADS is investigated. We also construct the difference scheme with the Euler-Maclaurin formula instead of the trapezoidal rule and investigate numerically that the error of the method equals O Let us consider the boundary-value problem where the boundary and initial conditions are coincidental with the exact solution u(x, t) = exp B 0 e t + 1 2 − x 2 4 , and the function Q(u) = u ln u is a source for u > 1 and a sink for 0 < u < 1 [12]. We approximate it by the difference scheme From (32) we find that x n+1 To solve equation (33) we use iteration method (30). The stopping criterion in the iteration method is   Now we investigate numerically stability of the considered scheme. We perturb the initial valuesỹ 0 Fig. 2).    To obtain better numerical results, under the condition f (u) ∈ C 2M +2 (R), where M = const > 0, we use the Euler-Maclaurin formula instead of the trapezoidal rule [2]  (−1) j (2M )! (2M −2j+1)! a j . We estimate a posteriori the error of this method [13] y n − y 2n Cn , x n i ∈ ω, x 2n i ∈ ω hτ /2 , Cn , x n i ∈ ω hτ /2 , x 2n i ∈ ω hτ /4 .
Here we use two additional grids ω hτ /2 , ω hτ /4 with the time step twice and four times smaller than τ , respectively. Tables 3 -4 present numerically that the error max

Conclusions
In the paper, we have considered the EDS for the Cauchy problem for the semilinear parabolic equation. The solution with the separated variables u(x, t) = X(x)T 1 (t) + T 2 (t) was a very important assumption. The ADSs have been constructed in the case when Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 06/04/2022 10:22:55 U M C S 108 Difference schemes of arbitrary order of accuracy for semilinear parabolic equations integrals in EDS cannot be evaluated exactly. The special attention was paid to investigate approximation, stability and convergence of the nonlinear scheme. Convergence of the iteration method was also considered. Numerical results have been presented to confirm the theoretical results given in the paper.