Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

1Department of Mathematics, University of Engineering and Technology, Lahore 54840, Pakistan 2Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, 22100 Como, Italy 3Departament de Fı́sica i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d'Urgell 187, 08036 Barcelona, Spain 4Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Heat conduction problems with suitable boundary conditions exist in many areas of engineering applications [1][2][3][4][5][6][7].Historically, highly accurate compact finite difference schemes are developed in the work by Lele [8].But these higher-order compact finite difference schemes only offer good accuracy at the interior nodes or for periodic boundary conditions.Usually, compact finite difference schemes have first-or second-order accuracy [9,10].The low-order of accuracy near boundary grid points affects the whole numerical results and it reduces the accuracy of overall numerical solution [11].Some authors offer one-side finite difference approximations for the Dirichlet boundary condition [8,12] but they cannot offer unconditional stability for the whole finite difference scheme.Recently, Dai et al. [11,[13][14][15][16] proposed a new idea to achieve higher-order accuracy with unconditional stability.
Actually, authors introduced a new parameter  that adjusts the location of nodes near the boundaries in symmetric way.
Han and Dai [17] have proposed a compact finite difference method for the spatial discretization of (1a) that has eighth-order accuracy at interior nodes and sixth-order accuracy for boundary nodes.Actually, they use eighthorder accurate approximation for the second-order derivative developed in [8] and given as where double dash always means derivative with respect to spatial variable and  3 = 344/1179,  3 = (38 3 − 9)/214,  3 = (696−1191 3 )/428, and  3 = (2454 3 −294)/535.The implicit compact finite difference scheme (4) without the contribution of boundary nodes in matrix form is given by Here, The governing equation of implicit compact finite difference approximation of second-order derivative is Note that (4) can be obtained from (7).We can observe that matrix  is strictly diagonally dominant and matrix  is diagonally dominant.In [17], authors constructed finite difference scheme at boundary nodes in such a way that they conserve the diagonal dominance of matrixes  and  to attain higher order of accuracy.The achieved order of accuracy at boundary nodes in [17] is six.The essence of article [17] is hidden in the calculation of a parameter  that they use to obtain higher-order approximation near boundary nodes.The parameter  makes the point distribution unequal near boundary but offer higher order of accuracy and diagonal dominance for matrices  and .The diagonally dominance of  and  is the key to prove stability of whole compact finite difference scheme when they use Crank-Nicholson method for time integration.
We construct a new finite difference scheme for boundary nodes to achieve eighth order of accuracy that exactly matches the order of accuracy at the interior nodes.The inclusion of two parameters  and 's will be introduced in our newly developed finite difference scheme for boundary nodes.The diagonally dominance will be conserved and the whole eighth-order scheme becomes stable.The stencil of eighth-order implicit finite difference scheme to approximate the second-order derivative for the interior nodes is { − 2,  − 1, ,  + 1,  + 2} for  = 3, 4, . . .,  − 2. (8) It means that if we are at location , then we need two grid nodes to the left of  and two grid points to the right of it.It is noticeable that mutual distance between nodes is equal to ℎ = /(2 +  − 2) and  points divide the interval [0, ] with  0 = 0,  +2 = ,  1 −  0 = ℎ,  +1 −   = ℎ, and   = ( +  − 1)ℎ for  = 2, 3, . . .,  − 1. Figure 1 shows the location of interior and boundary nodes.

Interior nodes
In the case of Dirichlet boundary conditions, we construct the eighth-order accurate implicit finite difference approximations of second-order derivatives for the nodes  1 ,  2 ,   , and  +1 , whereas, in the case of Neumann boundary conditions, we establish stable fourth-order compact finite difference scheme for the entire grid.

Compact Implicit Finite Difference Scheme for Boundary Nodes in the Case of Dirichlet Boundary Conditions
In the case of Dirichlet boundary condition, we construct the following compact implicit finite difference scheme at point  1 = ℎ: where ,  1 ( ̸ =0), and   for  = 0, 1, 2, . . ., 8 are unknowns and we are interested to find them in such a way that we can get (i) eighth-order accurate approximation of second-order derivative at  1 , (ii) diagonal dominance, that is,  1 < 0 and By expanding (9) around the node  1 = ℎ and equating the coefficients of the same-order derivatives, we get the following system of equations: By solving (10), we get the solution as follows: We define the error equation as After substituting the solution ( 11) into ( 12), we get where ⋅  (10)    2 = 53222400 (2 7 + 49 6 + 483 5 + 2450 4 Similarly, at the node  2 , we have the following construction of the implicit compact finite difference scheme: where ,  1 ( ̸ =0),  2 ( ̸ =0), and   for  = 0, 1, 2, . . ., 8 are unknowns and we are interested to find them in such a way that we can get (i) eighth-order accurate approximation of second-order derivative at  2 , (ii) diagonal dominance, that is,  2 < 0 and (15) around the node  2 = ( + 1)ℎ and equating the coefficients of the same-order derivative, we get the system of the following: By solving (16), we get the following solution: Advances in Numerical Analysis The error equation at the node  2 is given by By substituting ( 17) into (18), we get where Advances in Numerical Analysis 7 + 131725440)  (10)    2 = 479001600 (2 7 + 49 6 + 483 5 + 2450 4 In a similar way to that for the nodes  1 and  2 , the implicit compact finite difference approximation of second-order derivatives at nodes   and  −1 is constructed.

Implicit Compact Finite Difference Scheme for Boundary Nodes in the Case of Neumann Boundary Conditions
For the construction of fourth-order implicit compact finite difference approximation for second-order derivatives at node   ,  = 2, 3, . . ., , we consider the following model: The stencil in (21) is and its length is three.It means that we have to construct a fourth-order approximation of second-order derivative at nodes  1 and   .Due to symmetry, we will only present the construction for the node  1 in the case of Neumann boundary conditions.By expanding (21) around the node   and comparing the coefficients of the same-order derivative of  at   , we find the following values: and ( 21) becomes where  = 2, 3, . . ., .Equation ( 24) can be written as The error equation for ( 24) is given by Next, we consider the following model for the construction of implicit compact finite difference approximation of the second-order derivative at node  1 : After expanding (27) around  1 and simplifying it, we get the values of the parameters: where  = √ 225 + 30 √ 30.The error equation for the stencil ( 27) is given by )  (6)   ℎ 4 +  (ℎ 5 ) . (29)
Theorem 1. Prove that if matrices  and  are positive definite, then matrix  −1  is also positive definite in the sense that its eigenvalues have positive real parts.
In Table 1, we computed the spatial rate of convergence of our proposed implicit compact finite difference scheme the Dirichlet problem.In all cases, Table 1 shows that the spatial rate of convergence is at least eight.In Table 2, we measure the temporal rate of convergence of Crank-Nicholson method which is at least two and it is according to the theoretical rate of convergence.Similarly, in Tables 3  and 4, we computed the spatial and temporal rates, respectively, for Neumann problem and found them according to theoretical values.In the case of Dirichlet problem, the variation of  does not affect error in temporal dimension but does affect the error in spatial dimension and that is why Table 2 shows the same values of error with respect to different values of  and Δ.We also observe that as we decrease either ℎ, Δ, or both, we get reduction in the error (, Δ).Decreasing of ℎ means that we increase the number of grid points  in the spatial domain and this is also valid in temporal dimension.The temporal domain for the Dirichlet and Neumann problems is chosen to [0, 1] as the solution decays rapidly when the time passes 1.The maximum error in the numerical solution occurs around 0.1 in temporal dimension; this is the reason why we just integrate the Dirichlet problem in the temporal domain [0, 0.1] (see Table 1).

Conclusions
The implicit compact finite difference methods provide a more accurate way to approximate the spatial derivatives compared to explicit finite difference methods.The construction of compact finite difference operators for the interior nodes provides diagonal dominance and positivity of the diagonal entries which is in fact a very nice property which finally appears in the form of positive deftness of the compact operator.We have observed that the positive deftness helps us to prove the stability of numerical algorithm to solve 1D heat conduction equations.However, the diagonal dominance and positivity of diagonal entries for the interior nodes are not enough, because we also have to deal with the boundary conditions and usually one-sided compact finite difference schemes do not respect the nice property of positive definiteness with high order of convergence rate.In this project, the designing of the high-order accurate boundary conditions is established in such a way that we can maintain the positive definiteness of compact operator for the numerical scheme.

Figure 1 :
Figure 1: Grid points and location of boundary nodes.