A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation

To find a numerical solution for soliton-like structures model, we propose an adaptive meshless method based on the optimal sampling density (OSD) of kernel interpolation. We first consider the relationship between the optimal nodal distribution and the error bound of kernel interpolation, and obtain the corresponding OSD. Then we introduce an OSD based kernel interpolation method to approximate a function. And a numerical two-step meshless method is finally suggested for soliton-like structures model, taking the sine-Gordon equation as an example. In each time level, the predictor process takes field nodes with the same node distribution, while the final process takes field nodes arranged adaptively according to each OSD. With only a little added computational cost, the solution accuracy can be much improved. From the numerical examples, it is shown that the proposed method is very helpful for simulating soliton-like structures model.