Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation ()

Masaharu Nakashima^{}

Kagoshima-shi, Taniyama Chuou 1-4328 891-0141, Japan.

**DOI: **10.4236/jamp.2015.311176
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Kagoshima-shi, Taniyama Chuou 1-4328 891-0141, Japan.

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio *k/h*^{2} where *k* and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

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Nakashima, M. (2015) Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation. *Journal of Applied Mathematics and Physics*, **3**, 1506-1521. doi: 10.4236/jamp.2015.311176.

1. Introduction

A number of difference schemes for solving partial difference equations have been proposed by using the idea of

methods of lines [2] [3] . The scheme is required the condition of step size ratio for some constant,

where k and h are step sizes for space and time respectively. We [1] [4] -[6] have proposed some explicit scheme and overcome this problem. The problem considered in this paper is linear and nonlinear parabolic problem

(1.1)

with the initial Dirichlet boundary condition

(1.2)

where we set

(1.3)

We propose the difference approximation to (1.1) where the step size ratio is defined by

(c is any positive constant) (1.4)

The outline of this paper is as follows. In §2, by using idea of methods of lines, we present the explicit difference approximation to (1.1). In §3 we study the truncation errors of our scheme. In §4 we study the convergence of the scheme with the condition (1.4) and we will show that our scheme converges to the true solution of (1.1). In §5 we study stability of the scheme, and we will show that our scheme is stable for any step size k and h with the condition (1.4). In §6 we show some numerical examples to justify our methods.

2. Difference Scheme

In the same way as in [1] , we will approximate (1.1) by replacing the derivative for space and time in the difference operator

(2.1)

where is the central difference operator, forward difference operator. We denote the approximation to (1.1) at the mesh point

We set

(2.2)

We define the difference approximation to (1.1) by the following scheme.

If.

Then we set

(2.3)

If.

Then we set

(2.4)

where

(2.5)

The step size in (2.3), (2.4) is defined by

(2.6)

If we set

(2.7)

Then, from (2.3), (2.4), we have

(2.8)

3. Truncation Error

We define the truncation error of (2.8)

(3.1)

where, from the definition of (2.7), we have

(3.2)

By Taylor series expansions of the solution of (1.1), we have

(3.3)

From (3.3), we have

(3.4)

If we set

(3.5)

and

(3.6)

Then, from (3.4), we have the following result.

Theorem 1. The truncation error of the difference approximation (2.8) to (1.1) is given by

(3.7)

where

(3.8)

where and are defined by (3.5) and (3.6) respectively.

4. Convergence

In this section, we study the convergence of the scheme (2.8). We set the approximation error by

(4.1)

We use the abbreviation's

From (2.8), (3.7), (4.1), we have

(4.2)

with

(4.3)

From (2.5), we have

(4.4)

(4.5)

We set the initial conditions of (4.2)

(4.6)

Form (4.2), (4.4), (4.5), (4.6), we have

(4.7)

From (4.7), we have

(4.8)

with

(4.9)

(4.10)

We study the coefficients of (4.8) to.

Firstly we consider the case

(4.11)

We set

(4.12)

Then from (4.3), (4.12), we have

(4.13)

(4.14)

We have the equation

(4.15)

(4.16)

From (4.13), (4.14), (4.15), (4.16), we have

(4.17)

If we assume

(4.18)

Then we have

(4.19)

From (3.7), we have

(4.20)

with

(4.21)

From (4.20), we have

(4.22)

where is defined by (4.21).

We have from the condition (1.1)

(4.23)

From (4.17), (4.20), (4.23), we have

(4.24)

where is defined by (4.21) .

In the same way to (4.16), from (4.10), we have

(4.25)

From (3.8), we have

(4.26)

After some complicate computation, we have

(4.27)

with

From (4.27), we have

(4.28)

with

(4.29)

From (4.26), we have

(4.30)

with

From (4.30)

(4.31)

with

(4.32)

From (4.26), (4.28), (4.31), we have

(4.33)

From (4.25), we have

(4.34)

From (4.25), (4.33), (4.34), we have

(4.35)

where and are defined by (4.29) and (4.32) respectively.

From (4.20), we have

(4.36)

where is defined by (4.21).

From (4.8), (4.20) (4.24), (4.35), (4.36), we have

(4.37)

where and are defined by (4.21), (4.29) and (4.32) respectively.

We set the maximum norm of the function

(4.38)

Then, from (4.37), we have

(4.39)

From (4.39), we have

(4.40)

Finally we assume

(4.41)

Then, from (4.3), we have

(4.42)

From (4.9), (4.42), we have

(4.43)

In the same way to (4.14), we have

(4.44)

From (3.8), we have after some computation,

(4.45)

with

(4.46)

From (4.8), (4.20), (4.43), (4.44), (4.45), we have

(4.47)

where and are defined by (4.21) and (4.46) respectively.

Then, in the same way to (4.40), from (4.47), we have

(4.48)

We study l = 0. In the almost same way to (4.47), we have

(4.49)

where C_{1} and C_{4} are defined by (4.21) and (4.46) with l = 0 respectively.

From (4.49), we have

(4.50)

From (4.40), (4.48), (4.50), we have

Theorem 2. Suppose that for and, there exists positive numbers and such that

If the solution of (1.1) satisfies conditions (4.18). Then, the approximation generated by the scheme (2.8) converges to the solution of the differential Equation (1.1).

5. Stability

In this section, we study the stability of the numerical process (2.8) and define as follows.

Definition: The numerical processes is stable if there exists a positive constant such that

where denotes some norm and the constant is depends on initial value.

We prove that the scheme (2.8) are stable in mean of the von Neumann.

We set

Then, from (4.7), we have

(5.1)

From (5.1), we have

(5.2)

where and are defined by (4.9), (4.10) and (3.8) respectively.

If we assume (4.18) on the solution of (1.1), Then,in the same way to (4.31), (4.33), (4.45), we have

(5.3)

for some constant.

From (5.2), (5.3), we have the following result.

Lemma 1. If we assume the solution of (1.1) satisfies (4.18), Then there exists the constant such that

(5.4)

with (5.5)

where is defined by (5.3). From (2.8), we have

(5.6)

We set the maximum norm of the function

(5.7)

We have the inequality

(5.8)

From (1.1), we have

From (5.8), we have

(5.9)

From (2.8), we have

(5.10)

and

(5.11)

From (5.10), (5.11), we have

(5.12)

Firstly we consider

Then from (5.9) and (5.12), we have

(5.13)

with

(5.14)

where K, are defined by (4.19) and (5.5) respectively.

From (5.14), we have

(5.15)

Lastly, we consider

From (5.12), we have

(5.16)

Firstly, we consider the case.

Then from (5.16), we have

(5.17)

We have

(5.18)

From (5.10). (5.17), (5.18), we have

(5.19)

with

(5.20)

where K and are defined by (4.19) and (5.5) respectively.

If, Then we set

(5.21)

From (5.21), we have

(5.22)

If, Then we set

(5.23)

From (5.23), we have

(5.24)

From (5.22), (5.24), we set

(5.25)

where and are satisfy (5.21) and (5.23) respectively.

From (5.6), (5.19) and (5.25), we have

and we have the following result

(5.26)

From (5.26), we have

(5.27)

where is defined by (5.25).

Secondly, in the case, from (5.12), we have

(5.28)

From (5.28), we have

(5.29)

with

(5.30)

where K and are defined by (4.19)and (5.5) respectively.

In the same way to (5.26), we have

(5.31)

where is defined by (5.30).

From (5.15), (5.27), (5.31), we have

Theorem 3.

If the solution of (1.1) is analytic on the region then the difference approximation (2.8) to (1.1) are stable.

6. Numerical Example

Lastly, we study the numerical test in the following non-linear Equation .

(6.1)

and the initial and boundary problem given by,

(6.2)

Applying the difference Equation (2.8) to (6.1) with (6.2), we have the the numerical results in Table 1 and Figure 1, Figure 2.

Table 1. (x = 0/100, 2/100, 20/100, 50/100, 70/100, 98/100), (t = 0, 2/100, 10/100, 20/100, 50/100).

Figure 1. Initial data (0 ≤ x ≤ 1, t = 0).

Figure 2. The numerical solution for 0 ≤ x ≤ 1, t = 50.

As we see in Figure 1, Figure 2, the initial data diffuses slowly. Here the interval [0,1] is divided into

with.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
Nakashima, M. (2013) A Study on Unconditionally Stable Explicit Difference Schemes for the Variable Coefficients Parabolic Differential Equation. IJPAM, 87, 587-602. http://dx.doi.org/10.12732/ijpam.v87i4.8 |

[2] |
Du Fort, E.G. and Frankel, E.G. (1953) Stability Conditions in the Numerical Treatment on Parabolic Differential Equations. Mathematical Tables and Other Aids to Computation, 17, 135-152. http://dx.doi.org/10.2307/2002754 |

[3] | Schiesser, W.S. (1991) The Numerical Methods of Lines. Academic Press, San Diego. |

[4] | Nakashima, M. (2001) Unconditionally Stable Explicit Difference Schemes for the Variable Coefficients Parabolic Differential Equation (II). Processing Techniques and Applications International Conference, Las Vegas, June 2001, 561-569. |

[5] | Nakashima, M. (2002) Unconditionally Stable Explicit Difference Schemes for the Variable Coefficients Parabolic Differential Equation (IV). In: Dinov et al., Eds., Numerical Methods and Applications, Lecture Notes in Computer Science, Springer, Vol. 2542, 536-544. |

[6] | Nakashima, M. (2003) Unconditionally Stable Explicit Difference Schemes for the Variable Coefficients Two Dimensional Parabolic Differential Equation (V). Journal of Applied Mechanics, 54, 327-341. |

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