An adaptive meshfree method for nearly singular solutions of the KdV equations

An adaptive meshfree method for nearly singular solutions of the KdV equations

In this paper, we are interested in solving the one-dimensional nonlinear KdV equation, whose solution presents rapid variations or high gradients in some local regions of the domain. The radial basis function (RBF) collocation method is combined with the nite dierence scheme, because, unlike with Kansaʹs method, nonlinear PDEs can be converted to a system of linear equations, and a new adaptive algorithm is used to choose the location of the collocation points in any time step. Also, numerical stability can be obtained for our scheme, using the spectral norm of the amplication matrix. Results show that the method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution and the adaptive grading scheme is eective in localizing oscillations.