Mesh | |||||
4.4921e-2 | 1.1041e-2 | 2.8764e-3 | 7.1999e-4 | 1.6418e-4 | |
Rate | - | 2.0248 | 1.9405 | 1.9982 | 2.1327 |
5.8460e-2 | 1.3052e-2 | 3.2214e-3 | 8.0842e-4 | 1.9908e-4 | |
Rate | - | 2.1632 | 2.0185 | 1.9945 | 2.0218 |
A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.
Citation: |
Table 1.
Numerical results for the KPI equation obtained by the present scheme:
Mesh | |||||
4.4921e-2 | 1.1041e-2 | 2.8764e-3 | 7.1999e-4 | 1.6418e-4 | |
Rate | - | 2.0248 | 1.9405 | 1.9982 | 2.1327 |
5.8460e-2 | 1.3052e-2 | 3.2214e-3 | 8.0842e-4 | 1.9908e-4 | |
Rate | - | 2.1632 | 2.0185 | 1.9945 | 2.0218 |
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Left: the errors in solution (Solid line:
The lump type solitary wave at time
The lump type solitary wave at time
The profiles of interaction of the two lump waves at different times and the variation of the global energy (the last graph):
The motion of single soliton for the KPII equation. Top: