Parallel implementation of the split-step and the pseudospectral methods for solving higher KdV equation

doi:10.1016/S0378-4754(02)00188-X

Mathematics and Computers in Simulation

Volume 62, Issues 1–2, 13 February 2003, Pages 41-51

https://doi.org/10.1016/S0378-4754(02)00188-X Get rights and content

Abstract

Numerical simulations show that higher order KdV equation under certain conditions has a self-focusing singularity, which means that the solution of the equation blows up in finite time. In this paper, two numerical schemes: the split-step Fourier transform and the pseudospectral methods are used to investigate this self-focusing singularity problem. Parallel algorithms for the proposed schemes are designed and implemented. FFTW-MPI algorithm designed by Matteo Frigo and Steven Johnson is used for parallel implementation of the discrete Fourier transform (DFT). The parallel algorithms are implemented on an SGI Origin 2000 multiprocessor computer and experiments show that a considerable speedup is attained.

Introduction

The Korteweg–de Vries equation (KdV) [6] describes the behavior of certain types of waves occurring in a shallow canal in terms of a nonlinear differential equation. This equation has played a central role in the study of nonlinear phenomena, which has flourished so much since the 1970s [1].

It has been shown that the higher order nonlinear KdV equation: $u_{t} +u^{k} u_{x} +u_{xxx} =0, k>3$ has a self-focusing singularity [3], [8], which means that the solution of Eq. (1) blows up in finite time. Recently, there has been a lot of theoretical and numerical research in order to investigate this phenomenon (see [3] and the references therein). Numerical simulations of solutions of Eq. (1) confirm that its solitary-wave solutions are unstable if k>3, and in fact, that neighboring solutions emanating from smooth initial data appear to form singularities in finite time [2], [4]. Taha [8] designed an accurate numerical scheme based on the inverse scattering transform (IST) to solve this problem. In this paper, two numerical schemes: the split-step Fourier transform and the pseudospectral methods [4], [7] are used to investigate this self-focusing singularity problem.

In order for the singularity to be properly resolved, the mesh sizes in the directions of x and t have to be taken very small. Therefore, the implementation of the proposed numerical schemes on a serial computer requires a huge amount of computing time.

In this paper, two parallel algorithms for the above schemes are designed and implemented on an SGI Origin 2000 hypercube, and numerical results are discussed.

Section snippets

Numerical schemes

The split-step Fourier method introduced by Tappert [9] and the pseudospectral method introduced by Fornberg and Whitham [4] are originally designed for solving the KdV equation. However, the same ideas can also be used for solving higher KdV equation.

Data distribution

For the implementation of this algorithm, the potentials u and ũ are discretized by using N-element vectors, where N is the total number of spatial mesh points. Let P₀,P₁,…,P_m−1 denote the processors in a multiprocessor computer, then each processor P_i will have LN=N/m elements of u and ũ. The local arrays u[1,…,LN] and $u ̃ [1,…, LN]$ in processor P_i are corresponding to the elements $u[LN^{∗} i+1,…, LN^{∗} (i+1)]$ and $u ̃ [LN^{∗} i+1,…, LN^{∗} (i+1)]$ in the global array. Fig. 1 shows an example of data distribution

Results and conclusion

The proposed numerical scheme is implemented on an SGI Origin 2000 multiprocessor computer. Eq. (1) is solved at each time step for k=4.

According to our numerical experiments, the solution of Eq. (1) for k=4 blows up at t=0.147347 with Δx=0.019531 (N=4096) and Δt=0.0000072 for the split-step Fourier method, and at t=0.145648 with Δx=0.019531 (N=4096) and Δt=0.0000001 for the pseudospectral method. Fig. 3 shows the initial condition. Fig. 4, Fig. 5, Fig. 6, Fig. 7 show the solutions at t equals

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¹: Present address: Department of Computer and Information Science, The University of Michigan-Dearborn, Dearborn, MI 48128, USA.

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Parallel implementation of the split-step and the pseudospectral methods for solving higher KdV equation

Abstract

Introduction

Section snippets

Numerical schemes

Data distribution

Results and conclusion

Comput. Math. Appl. A

J. Comput. Phys.

Stability and instability of solitary waves of Korteweg–de Vries type

Proc. R. Soc. London, Ser. A