Parallel Algorithms For The KdV Equation And Application

In this paper, we constructed several parallel algorithms for solving the third-order KdVequation with initial condition and boundary condition.First，we give some basic difference schemes to approach the KdV equation，which includedthe classic explicit difference scheme, the classic implicit difference scheme and the Crank-Nicolsonscheme。In the same time, we also give the follow-flow scheme and two kinds of Saul’yevasymmetric difference schemes。Second，according to such schemes，we constructed four parallel algorithms。Using the firstkind of asymmetric difference schemes, we get the first kind of alternating group scheme based onfollow-flow scheme for the KdV equation; Using the classic explicit difference scheme, the classicimplicit difference scheme and the first kind of asymmetric difference schemes, we get thealternating segment explicit-implicit scheme based on follow-flow scheme for the KdV equation;Using the second kind of asymmetric difference schemes, we get the second kind of alternatinggroup scheme based on follow-flow scheme for the KdV equation; Using the Crank-Nicolsonscheme and the second kind of asymmetric difference schemes, we get the alternating segmentCrank-Nicolson scheme based on follow-flow scheme for the KdV equation;Next，using the matrix method，we discus the linear stability analysis. The theorem shows thatthe parallel algorithms constructed here are absolutely stable.Finally，some numerical experiments for the cases of single soliton solution and double solitonsolution are performed. We test the stability, the convergence order and the accuracy. Experimentsshow that these parallel algorithms constructed here have the intrinsic parallelism, absolute stabilityand high accuracy. In addition，we compare them with the existed schemes，the results show that theparallel algorithms constructed here are more accurate, it agrees with the previous theoretic analysis.