^{1}

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An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented.

We adopt the method of lines approach which is commonly used for solving time-dependent partial differential equations (PDE), whereby the spatial derivatives are replaced by finite difference approximations (see Lambert [

subject to the initial/boundary conditions

We seek a solution in the strip

We then define

occurring in (1) by the central difference approximation to obtain

discrete problem

which can be written in the form

where

The paper is organized as follows. In Section 2, we derive a continuous approximation which is used to obtain the BHSDA. The BHSDA is also analyzed in section 2. The computational aspects of the method is given in Section 3. Numerical examples are given in Section 4 to show the accuracy of the method. Finally, the conclusion of the paper is discussed in Section 5.

We begin by considering a scalar form of (3)

where we assume that the function f is Lipshitz continuous and the problem (4) possesses a unique solution. Furthermore, let

where

In order to uniquely determine the unknown coefficients

We note that (6) leads to a system of five equations which is solved by Cramer's Rule to obtain

where

method (7) is then evaluated at

Remark 2.1 In order to conveniently analyze and implement the method (8), we will express it in block form as given in (9).

where

Define the local truncation error of (4) as

where

operator. Assuming that

method is of order four.

Proposition 2.2 The BHSDA (9) applied to the test equations

with the amplification matrix

Remark 2.3 The dominant eigenvalue of

function called the stability function which determines the stability of the method.

Proof. We begin by applying (2) to the test equations

Definition 2.4 The block method (9) is said to be 1)

Corollary 2.5 The method (9) is

Proof: The dominant eigenvalue

proof follows from definition 2.4.

Remark 2.6 The stability region for the method (9) is given in

The resulting system of ODEs (3) is then solved on the partition

the grid index.

Step 1: Use the block method (9) to solve (3) on rectangles

Step 2: Let

and

Step 3: Step 2 is repeated for

mations

We note that for linear problems, we solve (3) directly with our Mathematica code enhanced by the feature

Computations were carried out in Mathematica 9.0 and the errors were calculated as

Example 4.1 As our first test example, we solve the given PDE (see Cash [

The exact solution

In

Example 4.2 As our second test example, we solve the given stiff parabolic equation (see Cash [

The exact solution

Cash [

Crank-Nicolson | Cash (2.6a, b) | Cash (2.13a, b, c) | BHSDA | |||
---|---|---|---|---|---|---|

1 | ||||||

1 | ||||||

5 | ||||||

5 |

BHSDA | Crank-Nicolson | Cash (2.6a, b) | Cash (2.13a, b, c) | |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

5 | ||||

10 |

We have proposed a BHSDA for solving parabolic PDEs via the method of lines. The method is shown to be

Fidele FouogangNgwane,Samuel NemseforJator, (2014) L-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations. American Journal of Computational Mathematics,04,87-92. doi: 10.4236/ajcm.2014.42008