Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(\tau^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis.


Introduction
Recently, a great deal of efforts have been devoted to the development of numerical approximation of heat equation problems (see, [1,3,8,9]). It is well known that the traditional numerical schemes have low accuracy, and thus need fine discretization to obtain desired accuracy, which will present many computational challenges due to the prohibitive computer memory and CPU time requirements (see, [3]).
For heat equations, the forward Euler methods, backward Euler methods and Crank-Nicolson methods were presented in [1]. In addition, three layer implicit schemes also appeared in Ref. [8]. The forward Euler method and backward method only have one-order accuracy in time and two-order accuracy in space. Also, the forward Euler method is not stable when cτ /h 2 ≤ 1/2. The three layer implicit compact format can reach O(τ 2 + h 4 ), but the format is complex. The Crank-Nicolson method has two-order accuracy in time and two-order accuracy in space, which is not good compared to the high-order compact difference scheme (see, [3]) with two-order accuracy in time and four-order accuracy in space. The high-order compact difference format (H-OCD) has many advantages, such as using less grid backplane point, high accuracy, good stability. This scheme plays more and more important role in the numerical solution of partial differential equations and the computational fluid mechanics field. But the calculation will be increased rapidly with the increase of grid points. In addition, in the experimental process for the large problems, we found that the changes of the truncation errors of high-order compact difference schemes are large at first, and then very small along with the increase of the grid points. This article will give a new numerical gradient method based on the collocation polynomial and Hermite interpolation to overcome the above problems on the high-order compact difference schemes. First, we obtain the intermediate points of the grids by cubic and bi-cubic Hermite interpolation. And then, based on these intermediate points, one can deduce new explicit schemes for the gradient of the discrete solutions of heat equations, which will greatly reduce the amount of calculation in the same accuracy with the high-order compact difference schemes.
The outline of the article is organized as follows. In Section 2, a linearized compact difference scheme is derived for one-dimensional heat equations. And then the numerical gradient method is presented and its convergence is analyzed in detail in Section 3. In addition, the local Hermit interpolation and refinement are also introduced in Section 3. Finally, some numerical results are reported in Section 4.

The High-Order Compact Difference
Firstly, for convenience, let us consider the following one-dimensional heat equation problem where T is a positive number. Denote Ω = (0, 1) × (0, T ). In addition, the solution u(x, t) is assumed to be sufficiently smooth and has the required continuous partial derivative.
Next, let us recall the linearized compact difference scheme, which has been introduced in Ref. [2].
In the sequel, we sometimes use the index pair (j, k) to represent the mesh point (x j , t k ).
In order to obtain the high-order compact difference schemes on the equation (1), let us firstly consider it at the point (x j , t k+ 1 Next, for g(x) = [g 0 (x), g 1 (x), ..., g N (x)], we introduce the operator β with the help of Lemma 2.1. We denote By the famous Taylor formula, we have and where r k j = 1 24 Noting the initial and boundary conditions in the equation (1), we obtain the following high-order compact difference scheme.

A Numerical Gradient Method Based on the Local Hermite Interpolation and Collocation Polynomial
The truncation errors of compact difference methods talked about in section 2 are O(τ 2 + h 4 ) in [2]. However, the calculation will be increased rapidly with the increase of grid points. In addition, in the experimental process for the large problems, we found that the changes of the truncation errors of high-order compact difference scheme are large at first, and then very small along with the increase of the grid points (see Table 2,3,4 in Section 4).
In order to deal with those problems, we will give a new numerical gradient method based on the collocation polynomial and Hermite interpolation.
Let U h be the vector space of the grid function on Ω τ h . The u h denotes the discrete solution satisfying the formula (7)-(9). Denote Our improvements are as follows: (1) First, get the values of points u T j by H-OCD scheme (7)-(9) in Section 2; (2) Based on (1), we obtain the formula of P j with the help of collocation polynomial; i.e, (3) Then, get the intermediate points based on the Hermite interpolation; i.e., u T Through the above improvements, one can deduce a new explicit numerical gradient scheme for the gradient term of the discrete solutions of heat equa-tions, which will greatly reduce the amount of calculation in the same accuracy with the high-order compact difference format.
Next, according to our improvement scheme, let us analysis the convergence orders of u and its partial derivative on the refinement and collocation parts in the internal [0, 1].

The Local Hermite Interpolation and Refinement
For convenience, we just consider Hermite cubic and bi-cubic interpolation function u H (x, T ) on the interval [x j , x j+1 ] ⊂ Ω h , and its vertexes are as follows: On the segment z 1 − z 2 , let the cubic interpolation function satisfy the condition Based on [5], we can get the Hermite interpolation polynomial as follows where j = 1, 2, ..., N − 1. The interpolation errors are where ξ j is between z 1 and z 2 (see, [5]). So we have, by (11), the refinement computation format From (13), we know that the refinement schemes also have the four-order accuracy in terms of spacial.

The Collocation Polynomial
From the above analysis, we know that we must obtain the express of P j in order to get the specific formula of the intermediate points. Here, we choose the collocation polynomial method. For convenience, we just consider the sub-domain Then, we denote In order to get the approximation polynomial of u, we consider the polynomial space Because T are freely chosen, we can just think about the approximation polynomial of u when t = T . So we denote t = Mτ = T , then Then we can obtain the following numerical gradients by means of the collocation polynomial.
Next, let us analyze the accuracy of P j . For convenience, we first introduce some basic operators as follows: Let n be a positive number, and h = 1/(n − 1), then the region [a, b] can be discretized as Ω h = {x j |x j = a + jh, j = 0, 1, 2, ... n − 1}.
• The shift operator: We suppose that the arbitrary order derivatives of u(x) is existing, then using Taylor expansion to shift operators, we get So the shift operator can be written as • The inverse shift operator: Thus, we get the following theorems: x (Ω) and u(x, t) ∈ C 3 t (Ω), then we have where j = 2, 3, 4, ... N.
Proof. First, we use the Taylor expansion to shift operators, and we get Then Thus the proof is completed.2 Through the above theorem, we know that the accuracy of the partial derivative of u (i.e., P j ) is O(h 4 ) when t = T . In fact, due to (14), it is easy to prove that the accuracy of the intermediate points is O(h 4 ), too. The analysis is as follows.
Proof. First, we use the Taylor expansion to shift operators, and we get and then with the help of (14) and (17), we can make the derivations as follows: The theorem has been proved. 2 In addition, according to the constitution of P j and u T j+ 1 2 , they are obviously two-order in terms of time, so we do not talk about the time term in details here. Next, let us compare the different numerical solutions in the same number of grid points (note that the numerical gradient scheme need only the half of these grid points before the interpolation) with the exact solution as follows. Error( h 2 ) ), and Error(h) = max x k =x 0 +kh,k=0,1,...N {| (uxt(x k , T ) − u(x k , T )) |}, uxt(x k , T ) represents the exact solution and u(x k , T ) is the numerical solution.) and Error(P ) = max x k =x 0 +kh,k=0,1,...N {| (P (x k , T ) − P k ) |} with different temporal step sizes when spatial step size is fixed as h = 1/10000. In order to make sure that the dominated error is from temporal discretization, we use a very small h. From the table, we can draw the conclusion that the convergence orders of backward methods, Crank-Nicolson methods, the compact difference methods and the intermediate points (see Eq. (14)) which are obtained by the numerical gradient scheme in temporal are all O(τ 2 ). Table 2 lists the errors for a small and fixed τ = 1/100000, where h is different. The reason why we use a very small τ is to make sure that the dominated error is from spatial discretization. Obviously, the convergence orders in space are O(h 4 ). Table 3 lists the computational results of the intermediate points and u x with different temporal step sizes when spatial step size is fixed as h = 1/10000. We can see that the convergence orders in time can also reach O(τ 2 ). Table 1 Errors and rate of backward, C-N, H-OCD scheme (7)(8)(9)     Through these figures, one knows that combination of the compact difference and numerical gradient method in sparse grids is more better than other methods. Table 2 Errors and rate of backward, C-N and H-OCD scheme (7)(8)(9) in space direction with τ = 1/100000.    Next, let us compare the numerical solution with the exact solution as follows.

Numerical Experiments
From Tables 4 and 5, we know that the numerical results are consistent to our theoretical results.  In fact, if we use the Richardson extrapolation [10], we can get that the errors of H-OCD scheme (7)-(9) are four-order in terms of time. However, by Table 6, we conclude that the Richardson extrapolation is good for Problem 4.1, while it is worse for Problem 4.2. That is to say, the Richardson extrapolation method has some limitations. But, according to the constitute forms of the

Further work
Finally, the numerical gradient method can also be applied to the two-dimension heat equation ) can be expressed by the values u of the points and their partial derivatives P around it (see Fig. 6). P (x i , y j ) is connected to the points u around u(x i , y j ) (see Fig. 7). . For this problem, we will research in detail in the future.

Conclusions
Recently, many people devote themselves on the development of numerical approximation of heat equation problems. By the numerical comparisons, we can say that the compact difference scheme is better than the traditional numerical schemes. However, when the space step size h is too small, it will take longer time to get the approximate solution of problem. Also, the truncation errors of the compact difference method changes little along with the change of h. Fortunately, the method talked in this article can solve the problem to some extent. And from the last experiment, we know the Richardson extrapolation has some limitations. So we may choose it according to the practical problems. Finally, the numerical experiments showed that our theoretical analysis is true.