A Class Of Parallel Algorithms For The Generalized KdV Equation

The generalized KdV equation appears in many physical models and it occupies a quite important position in the physical problems of the nonlinear wave and the soliton theory. In recent years, more and more experts begin studying the numerical solving methods for the generalized KdV. We know that the classic explicit difference scheme, which is simple and able to be used for parallel computers straightly, often needs some strict stable conditions, the classic implicit scheme and Crank-Nicolson scheme is unconditionally stable, but they can't be used for parallel computation directly. With the coming of the information age and the development of the computers, parallel computation attracts more and more concerns for its character of solving the large and complicated computing problems rapidly. So, how to find a stable numerical solving method, which can be used for parallel computers directly, becomes an important problem needed to be solved as quickly as possible.The author provides a class unconditionally stable and parallelizable algorithms for the generalized KdV equation with period boundary condition.The author gives out the classic implicit scheme of the generalized KdV equation. on the base of the classic implicit scheme, we give out a parallelizable iterative algorithm for the generalized KdV equation. This parallel algorithm is unconditionally stability and can be used for parallel computers directly. Numerical examples show that the theoretical results are conformed to the numetical simulation.The author provides four kinds of Saulyev asymmetrical difference schemes to solve the generalized KdV equation. According to the above four Saulyev asymmetrical difference schemes, the author gives out the relevant numerical solving algorithms. They are the alternating 12-points group algorithm and a alternating segment Crank-Nicolson algorithm. The two algotithms are not only unconditionally stable but also have the parallel nature. Besides, our truncation error analysis and numerical experiment show that the numerical solution from the two algorithms all have a two-order rate of convergence in space.