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

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.

• Runge-Kutta Methods
• Method of Lines
• Difference Equation
• Non-Linear PDE

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

with the initial Dirichlet boundary condition

where we set

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

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

We set

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

If.

Then we set

If.

Then we set

where

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

If we set

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

3. Truncation Error

We define the truncation error of (2.8)

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

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

From (3.3), we have

If we set

and

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

where

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

We use the abbreviation's

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

with

From (2.5), we have

We set the initial conditions of (4.2)

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

From (4.7), we have

with

We study the coefficients of (4.8) to.

Firstly we consider the case

We set

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

We have the equation

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

If we assume

Then we have

From (3.7), we have

with

From (4.20), we have

where is defined by (4.21).

We have from the condition (1.1)

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

where is defined by (4.21) .

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

From (3.8), we have

After some complicate computation, we have

with

From (4.27), we have

with

From (4.26), we have

with

From (4.30)

with

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

From (4.25), we have

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

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

From (4.20), we have

where is defined by (4.21).

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

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

We set the maximum norm of the function

Then, from (4.37), we have

From (4.39), we have

Finally we assume

Then, from (4.3), we have

From (4.9), (4.42), we have

In the same way to (4.14), we have

From (3.8), we have after some computation,

with

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

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

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

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

where C₁ and C₄ are defined by (4.21) and (4.46) with l = 0 respectively.

From (4.49), we have

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

From (5.1), we have

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

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

with (5.5)

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

We set the maximum norm of the function

We have the inequality

From (1.1), we have

From (5.8), we have

From (2.8), we have

and

From (5.10), (5.11), we have

Firstly we consider

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

with

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

From (5.14), we have

Lastly, we consider

From (5.12), we have

Firstly, we consider the case.

Then from (5.16), we have

We have

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

with

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

If, Then we set

From (5.21), we have

If, Then we set

From (5.23), we have

From (5.22), (5.24), we set

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

From (5.26), we have

where is defined by (5.25).

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

From (5.28), we have

with

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

In the same way to (5.26), we have

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 .

and the initial and boundary problem given by,

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.

References

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. ,et al. (2003)Unconditionally Stable Explicit Difference Schemes for the Variable Coefficients Two Dimensional Parabolic Differential Equation (V) Journal of Applied Mechanics 54, 327-341.