## Second-generation wavelet collocation method for the solution of partial differential equations.(English)Zbl 0984.65105

A general framework for constructing accurate and efficient numerical methods of collocation type for solving nonlinear partial differential equations of the form $${{\partial u}\over{\partial t}}=F(x,t,u,\nabla u)$$ with boundary conditions (and possibly constraints) is developed. The considered methods are based on second-generation wavelets – in the paper lifted interpolating wavelets are applied.
The grid of collocation points is adapted dynamically with respect to time and reflects local changes in the solution. It is achieved by applying wavelet decompositions. Moreover, a new hierarchical finite difference scheme is described for calculating spatial derivatives of a function on an adaptive grid.
The proposed numerical method is applied to solving one-dimensional Burgers and modified Burgers equations and the one-dimensional diffusion flame problem. The numerical results indicate efficient adaptivity of the computational grid and associated wavelets to the local irregularities of the solution.

### MSC:

 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65T60 Numerical methods for wavelets 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
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