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An Alternating Group Explicit (AGE) iterative method with intrinsic parallelism is constructed based on an implicit scheme for the Regularized Long-Wave (RLW) equation. The method can be used for the iteration solution of a general tridiagonal system of equations with diagonal dominance. It is not only easy to implement, but also can directly carry out parallel computation. Convergence results are obtained by analysing the linear system. Numerical experiments show that the theory is accurate and the scheme is valid and reliable.

The Regularized Long-Wave (RLW) equation:

u t + u x + u u x − μ u x x t = f ( x , t ) , (1)

where μ > 0 .

It is a different explanation of nonlinear dispersive waves compared to the famous Korteweg-de Vries (KdV) equation, which has the form:

u t + u x + u u x + u x x x = f ( x , t ) . (2)

In the study of physical phenomena such as water wave and plasma wave propagation, the RLW equation is regarded as the modified model of the KdV equation. Compared with the KdV equation, the RLW equation has better mathematical properties and has been widely studied.

The RLW equation was first proposed by Peregrine [

With the rapid development of the high-performance computing in large-scale scientific and engineering computations, the parallel difference methods for partial differential equations have been studied rapidly. Evans [

Zhang and Liang [

The remainder of this paper is arranged as follows. In Section 2, we construct an AGE iterative method for RLW equation. The idea is that the computing process is designed as a number of small-size independent linear systems, which can be computed independently. In Section 3, we give the analysis of convergence, which is the main content of this paper. Finally, some numerical experiments show the effectiveness of our theoretical results.

Consider the following initial-boundary value problem for the RLW equation:

u t + u x + u u x − u x x t = 0 , ( x , t ) ∈ [ x L , x R ] × [ 0 , T ] , (3)

u ( x L , t ) = u ( x R , t ) = 0 , t ∈ [ 0 , T ] , (4)

u ( x , 0 ) = u 0 ( x ) , x ∈ [ x L , x R ] . (5)

where u 0 ( x ) is the known initial function.

Firstly, the computational region x j = j h ( 0 ≤ j ≤ J ) is meshed as follows. Let h and t be the space step and time step. Denote x j = j h ( 0 ≤ j ≤ J ) , t n = n τ ( 0 ≤ n ≤ N ) , u j n ≈ u ( x j , t n ) , where u ( x , t ) represents the exact solution, u j n represents the numerical solution.

The implicit scheme of the RLW Equation (3) is given as follows:

1 τ ( u j n + 1 − u j n ) + 1 4 h ( u j + 1 n + 1 − u j − 1 n + 1 + u j + 1 n − u j − 1 n ) + a j n 4 h ( u j + 1 n + 1 − u j − 1 n + 1 + u j + 1 n − u j − 1 n ) = 1 τ h 2 ( u j + 1 n + 1 − 2 u j n + 1 + u j − 1 n + 1 − u j + 1 n + 2 u j n − u j − 1 n ) , (6)

i.e.

( r 1 − r 3 ) u j − 1 n + 1 + 2 r 2 u j n + 1 + ( r 1 + r 3 ) u j + 1 n + 1 = ( r 1 + r 3 ) u j − 1 n + 2 r 2 u j n + ( r 1 − r 3 ) u j + 1 n , (7)

where r 1 = − 2 , r 2 = 2 + h 2 , r 3 = τ h 2 ( 1 + a j n ) , a j n = 1 2 ( u j + 1 n + u j − 1 n ) .

The linear system of the scheme (7) is as following:

A U = F , (8)

where

A = [ 2 r 2 r 1 + r 3 r 1 − r 3 2 r 2 r 1 + r 3 ⋱ ⋱ ⋱ r 1 − r 3 2 r 2 r 1 + r 3 r 1 − r 3 2 r 2 ] ( J − 1 ) × ( J − 1 ) ,

and U = [ u 1 n + 1 , u 2 n + 1 , ⋯ , u J − 1 n + 1 ] T , F = [ f 1 n , f 2 n , ⋯ , f J − 1 n ] T ,

f 1 n = ( r 1 + r 3 ) u 0 n + 2 r 2 u 1 n + ( r 1 − r 3 ) ( u 2 n − u 0 n + 1 ) ,

f j n = ( r 1 + r 3 ) u j − 1 n + 2 r 2 u j n + ( r 1 − r 3 ) u j + 1 n , j = 2 , 3 , ⋯ , J − 2 ,

f J − 1 n = ( r 1 + r 3 ) ( u J − 2 n − u J n + 1 ) + 2 r 2 u J − 1 n + ( r 1 − r 3 ) u J n , u 0 n + 1 = u J n + 1 = 0 , n = 0 , 1 , 2 , ⋯ .

Split the matrix A, we have:

A = G 1 + G 2 , (9)

where

G 1 = [ r 2 P P ⋱ P ] ( J − 1 ) × ( J − 1 ) , G 2 = [ P P ⋱ P r 2 ] ( J − 1 ) × ( J − 1 ) ,

and the block submatrix is:

P = [ r 2 r 1 + r 3 r 1 − r 3 r 2 ] .

Then, the AGE iterative method is constructed as follows:

{ ( I + G 1 ) U ( k + 1 2 ) = ( I − G 2 ) U ( k ) + F , ( I + G 2 ) U ( k + 1 ) = ( I − G 1 ) U ( k + 1 2 ) + F , k = 0 , 1 , 2 , ⋯ . (10)

Remark 1. The AGE iteration method (10) is a linear system, and the coefficient matrix is quasi-diagonal matrix. This matrix can be divided into several sub-block linear equations systems and calculated independently. Therefore, scheme (10) can do parallel processing calculations.

Remark 2. Obviously, we obtained that G 1 and G 2 are strictly dominance matrices, i.e.

r 2 − r 1 + r 3 = 4 + h 2 + τ h 2 ( 1 + a j n ) > 0 , (11)

r 2 − r 1 − r 3 = 4 + h 2 − τ h 2 ( 1 + a j n ) > 0. (12)

By Gershgorin circle theorem, it follows that G 1 and G 2 are positive definite matrices.

In this section, we will discuss the convergence of the AGE iterative method (10). The proof of convergence relies on the following Kellogg lemmas [

Lemma 1. If γ > 0 and C + C T is nonnegative definite, then ( γ I + C ) − 1 exists and

‖ ( γ I + C ) − 1 ‖ 2 ≤ γ − 1 .

Lemma 2. Under the conditions of Lemma 1, there is:

‖ ( γ I − C ) ( γ I + C ) − 1 ‖ 2 ≤ 1.

Theorem 1. The AGE iterative method (10) is convergent.

Proof. By eliminating U ( l + 1 2 ) ( l = 1 , 2 , ⋯ , k ) from (10), we can obtain:

U ( k + 1 ) = T U ( k ) + D k = ⋯ = T k + 1 U ( 0 ) + D 0 , (13)

where T = ( I + G 2 ) − 1 ( I − G 1 ) ( I + G 1 ) − 1 ( I − G 2 ) .

Let

T ^ = ( I + G 2 ) T ( I + G 2 ) − 1 = ( I − G 1 ) ( I + G 1 ) − 1 ( I − G 2 ) ( I + G 2 ) − 1 . (14)

From Remark 2, it is easily obtain that the matrices G 1 and G 2 are nonnegative real matrices.

Therefore, we can obtain the following inequalities from the above Kellogg lemmas:

ρ ( T ) ≤ ‖ T ‖ 2 = ‖ T ^ ‖ 2 ≤ ‖ ( I − G 1 ) ( I + G 1 ) − 1 ‖ 2 ‖ ( I − G 2 ) ( I + G 2 ) − 1 ‖ 2 = max 1 ≤ i ≤ m | 1 − α i 1 + α i | ⋅ max 1 ≤ i ≤ m | 1 − β i 1 + β i | < 1 , (15)

then

ρ ( T k ) ≤ ‖ T k ‖ 2 ≤ ( ‖ T ‖ 2 ) k < 1 , (16)

where α i , β i > 0 are the eigenvalues of the positive definite matrices G 1 and G 2 , respectively. We complete the proof.

In order to demonstrate the effectiveness and applicability of the proposed AGE iterative method (10) in this paper, we consider the following example:

u t + u x + u u x − u x x t = 0 , (17)

u ( x L , t ) = u ( x R , t ) = 0 , t ∈ [ 0 , T ] , (18)

u ( x , 0 ) = u 0 ( x ) , x ∈ [ x L , x R ] , (19)

where

− x L = x R = 50 , u 0 ( x ) = sech 2 ( x 4 ) . (20)

The single solitary wave solution of (17)-(19) is:

u ( x , t ) = A sech 2 ( k x − w t ) , (21)

where

A = 3 a 2 1 − a 2 , k = a 2 , w = a 2 ( 1 − a 2 ) , (22)

where a is arbitrary constant. This example takes a = 1 2 .

Taking τ = 1 / 10000 and h = 1 / 80 , we compare the errors among scheme (10), C-N scheme and other two algorithms in [

In

Scheme (10) | Shao [ | C-N scheme | Cai [ | |
---|---|---|---|---|

t = 0.2 | 1.575e−5 | 0.00056 | 0.00070 | 0.00053 |

t = 0.4 | 2.625e−5 | 0.00085 | 0.03331 | 0.00113 |

t = 0.6 | 6.728e−5 | 0.00112 | 0.06337 | 0.00175 |

t = 0.8 | 1.925e−4 | 0.00141 | 0.08433 | 0.00237 |

K | Scheme (10) | Cai [ | Shao [ | |
---|---|---|---|---|

h = 1/100 | 2 | 9.5615 s | 16.2625 s | 15.3531 s |

h = 1/400 | 8 | 5.0932 s | 33.4592 s | 32.0037 s |

h = 1/800 | 16 | 5.0946 s | 59.3013 s | 56.1752 s |

h = 1/1600 | 32 | 5.5882 s | 145.3313 s | 132.8453 s |

Since the scheme (10) is designed based on the implicit scheme (7), it obviously obtains the second-order spatial accuracy, i.e. O ( τ + h 2 ) . From

It is shown that the transmission process of single solitary wave from t = 0 to t = 20 with a spatial step size of h = 0.1 and h = 0.2 in

The numerical example results above present that the AGE iterative method (10) proposed in this paper has good computational accuracy and parallel efficiency with the same mesh division. Additionally, it can effectively simulate the transmission of single solitary wave, which demonstrates the applicability of our algorithm.

This work was supported by the National Natural Science Foundation of China (grant number 12101536) and the LCP Fund for Young Scholars (grant number 6142A05QN22004).

The authors declare no conflicts of interest regarding the publication of this paper.

Xie, A.Q., Ye, X.J. and Xue, G.Y. (2024) The Alternating Group Explicit Iterative Method for the Regularized Long-Wave Equation. Journal of Applied Mathematics and Physics, 12, 52-59. https://doi.org/10.4236/jamp.2024.121006