Abstract
Combining the variational iteration method (VIM) with the sparse grid theory, a dynamic sparse grid approach for nonlinear PDEs is proposed in this paper. In this method, a multilevel interpolation operator is constructed based on the sparse grids theory firstly. The operator is based on the linear combination of the basic functions and independent of them. Second, by means of the precise integration method (PIM), the VIM is developed to solve the nonlinear system of ODEs which is obtained from the discretization of the PDEs. In addition, a dynamic choice scheme on both of the inner and external grid points is proposed. It is different from the traditional interval wavelet collocation method in which the choice of both of the inner and external grid points is dynamic. The numerical experiments show that our method is better than the traditional wavelet collocation method, especially in solving the PDEs with the Nuemann boundary conditions.
Highlights
The sparse representation of functions via a linear combination of a small number of basic functions has recently received a lot of attention in several mathematical fields such as approximation theory as well as signal and image processing [1]
The interpolation wavelet such as the Shannon wavelet, Shannon-Gabor wavelet, Harr wavelet, and the autocorrelation function of the Daubechies scaling function can be taken as the basis function to construct the sparse grid approach directly
Faber-Schauder scaling function has no second-order derivative, it still can be the basis employed in the multiscale interpolation operator to solve the Burgers equation, while only retaining important nodes
Summary
The sparse representation of functions via a linear combination of a small number of basic functions has recently received a lot of attention in several mathematical fields such as approximation theory as well as signal and image processing [1]. The interpolation wavelet such as the Shannon wavelet, Shannon-Gabor wavelet, Harr wavelet, and the autocorrelation function of the Daubechies scaling function can be taken as the basis function to construct the sparse grid approach directly. The autocorrelation function of Daubechies scaling function has been widely used in various numerical methods for PDEs such as the wavelet collocation method and the sparse grids method. Daubechies scaling function has no exact analytical expression This will bring error to the approximation solution from the Daubechies wavelet numerical method. Compared with the finite difference method, the retained grid points are sparse and the dimensionality of the system of ODEs is smaller This is helpful to improve the efficiency, but the small change of the condition number and the smoothness of the function to be approximated can destroy the exactness of the numerical solution obtained by the traditional difference method. The last one is to construct a dynamic choice scheme on the external grid points, so that both of the inner and external grid points are dynamic with the development of the solution, especially to PDEs with the Neumann boundary conditions
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